Jim Douglas

Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time-stepping procedure such as Crank–Nicolson, can have errors associated with the AD or … Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time-stepping procedure such as Crank–Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time-stepping procedure. We show that minor modifications in the AD and FS procedures can virtually eliminate the perturbation errors at an additional computational cost that is less than 10% of the cost of the original AD or FS method. Moreover, after these modifications, the AD and FS procedures produce identical approximations of the solution of the differential problem. It is also shown that the same perturbation of the Crank–Nicolson procedure can be obtained with AD and FS methods associated with the backward Euler time-stepping scheme. An application of the same concept is presented for second-order wave equations.

NUMERICAL EXPERIMENTS FOR A NONOVERLAPPING DOMAIN DECOMPOSITION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

1996-05-01

Jim Douglas , Daoqi Yang

APPROXIMATION OF SCALAR WAVES IN THE SPACE-FREQUENCY DOMAIN

1994-08-01

Jim Douglas , Dongwoo Sheen , Juan E. Santos

A numerical method for approximating a pseudodifferential system describing attenuated, scalar waves is introduced and analyzed. Analytic properties of the solutions of the pseudodifferential systems are determined and used to … A numerical method for approximating a pseudodifferential system describing attenuated, scalar waves is introduced and analyzed. Analytic properties of the solutions of the pseudodifferential systems are determined and used to show convergence of the numerical method. Experiments using the method are reported.

An alternating-direction iteration method for Helmholtz problems

1993-01-01

Jim Douglas , Jeffrey L. Hensley , Jean E. Roberts

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Global estimates for mixed methods for second order elliptic equations

1985-01-01

Jim Douglas , Jean E. Roberts

Global error estimates in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^2}(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^\infty }(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript negative s Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^{ - s}}(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {R}}^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {R}}^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L p equals minus d i v left-parenthesis a bold g bold r bold a bold d p plus bold b p right-parenthesis plus c p"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi>div</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">g</mml:mi> <mml:mi mathvariant="bold">r</mml:mi> <mml:mi mathvariant="bold">a</mml:mi> <mml:mi mathvariant="bold">d</mml:mi> </mml:mrow> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">b</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Lp = - \operatorname {div}(a\;{\mathbf {grad}}\;p + {\mathbf {b}}p) + cp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> based on the Raviart-Thomas-Nedelec space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper V Subscript h Baseline times upper W Subscript h subset-of bold upper H left-parenthesis d i v semicolon normal upper Omega right-parenthesis times upper L squared left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">V</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>×</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>⊂</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>div</mml:mi> <mml:mo>;</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>×</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {V}}_h} \times {W_h} \subset {\mathbf {H}}(\operatorname {div};\Omega ) \times {L^2}(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Optimal order estimates are obtained for the approximation of <italic>p</italic> and the associated velocity field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold u equals minus left-parenthesis a bold g bold r bold a bold d p plus bold b p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">g</mml:mi> <mml:mi mathvariant="bold">r</mml:mi> <mml:mi mathvariant="bold">a</mml:mi> <mml:mi mathvariant="bold">d</mml:mi> </mml:mrow> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">b</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {u}} = - (a\;{\mathbf {grad}}\;p + {\mathbf {b}}p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^2}(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript negative s Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^{ - s}}(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-slanted-equals s less-than-or-slanted-equals k plus 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>s</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0 \leqslant s \leqslant k + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of bold upper R squared"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \subset {{\mathbf {R}}^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <italic>p</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^\infty }(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Interior and superconvergence estimates for a primal hybrid finite element method for second order elliptic problems

1985-01-01

Jim Douglas , Chaitan P. Gupta , Guangyu Li

Finite Difference Methods for Two-Phase Incompressible Flow in Porous Media

1983-08-01

Jim Douglas

Two-phase, incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Convection physically dominates diffusion, and the object of this paper is to develop a … Two-phase, incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Convection physically dominates diffusion, and the object of this paper is to develop a finite difference procedure that reflects this dominance. The pressure equation, which is elliptic in appearance, is discretized by a standard five-point difference method. The concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms. A convergence analysis is given for the method.

Numerical methods for a model of cardiac muscle contraction

1983-06-01

Jim Douglas , Fabio Milner

Numerical Methods for Convection-Dominated Diffusion Problems Based on Combining the Method of Characteristics with Finite Element or Finite Difference Procedures

1982-10-01

Jim Douglas , Thomas F. Russell

Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form $cu_t + bu_x - (au_x )_x = f$. Optimal … Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form $cu_t + bu_x - (au_x )_x = f$. Optimal order error estimates in $L^2 $ and $W^{1,2} $ are derived for the finite element procedure. Various error estimates are presented for a variety of finite difference methods. The estimates show that, for convection-dominated problems $(b \gg a)$, these schemes have much smaller time-truncation errors than those of standard methods. Extensions to n-space variables and time-dependent or nonlinear coefficients are indicated, along with applications of the concepts to certain problems described by systems of differential equations.

Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable

1981-01-01

Douglas N. Arnold , Jim Douglas , Vidar Thomée

A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal … A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal superconvergence is demonstrated. Discretization in time by explicit single-step methods is discussed.

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Superconvergence of a Finite Element Approximation to the Solution of a Sobolev Equation in a Single Space Variable

1981-01-01

Douglas N. Arnold , Jim Douglas , Vidar Thomée

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Superconvergence of the galerkin approximation of a quasilinear parabolic equation in a single space variable

1979-12-01

Douglas N. Arnold , Jim Douglas

Incomplete Iteration for Time-Stepping a Galerkin Method for a Quasilinear Parabolic Problem

1979-06-01

Jim Douglas , Todd Dupont , Richard E. Ewing

An iterative method is presented and analyzed which is based on using a preconditioned conjugate gradient iteration for approximately solving the linear equations produced at each time step by an … An iterative method is presented and analyzed which is based on using a preconditioned conjugate gradient iteration for approximately solving the linear equations produced at each time step by an extrapolated Crank–Nicolson–Galerkin procedure for time-stepping a quasilinear parabolic problem. Optimal order convergence rates are obtained for the iterative method which is shown to be (asymptotically) computationally more efficient than standard second-order-in-time correct methods.

A Quasi-Projection Analysis of Galerkin Methods for Parabolic and Hyperbolic Equations

1978-04-01

Jim Douglas , Todd Dupont , Mary F. Wheeler

Superconvergence phenomena are demonstrated for Galerkin approximations of solutions of second order parabolic and hyperbolic problems in a single space variable. An asymptotic expansion of the Galerkin solution is used … Superconvergence phenomena are demonstrated for Galerkin approximations of solutions of second order parabolic and hyperbolic problems in a single space variable. An asymptotic expansion of the Galerkin solution is used to derive these results and, in addition, to show optimal order error estimates in Sobolev spaces of negative index in multiple dimensions.

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A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations

1978-01-01

Jim Douglas , Todd Dupont , Mary F. Wheeler

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Local $H^{-1}$ Galerkin and adjoint local $H^{-1}$ Galerkin procedures for elliptic equations

1977-01-01

Jim Douglas , Todd Dupont , H.H. Rachford +1 more

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A Galerkin Method for a Nonlinear Dirichlet Problem

1975-07-01

Jim Douglas , Todd Dupont

A Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation $\nabla \cdot (a(x,u)\nabla u) = f$. … A Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation $\nabla \cdot (a(x,u)\nabla u) = f$. The asymptotic error estimates are of the same form as in the linear case. Newtonâs method can be used to solve the nonlinear algebraic equations.

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Optimal L ∞ Error Estimates for Galerkin Approximations to Solutions of Two-Point Boundary Value Problems

1975-04-01

Jim Douglas , Todd Dupont , Lars B. Wahlbin

The Effect of Interpolating the Coefficients in Nonlinear Parabolic Galerkin Procedures

1975-04-01

Jim Douglas , Todd Dupont

Error estimates are derived for a class of Galerkin methods for a quasilinear parabolic equation. In these Galerkin methods, both continuous and discrete in time, the nonlinear coefficient in the … Error estimates are derived for a class of Galerkin methods for a quasilinear parabolic equation. In these Galerkin methods, both continuous and discrete in time, the nonlinear coefficient in the differential equation is interpolated into a finite-dimensional function space in order to compute the integrals involved. Asymptotic error estimates of optimal order are produced.

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Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems

1975-01-01

Jim Douglas , Todd Dupont , Lars B. Wahlbin

A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary value problems. The class of Galerkin spaces considered includes almost all (quasiuniform) … A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary value problems. The class of Galerkin spaces considered includes almost all (quasiuniform) piecewise-polynomial spaces that are used in practice. The estimates are optimal in the sense that no better rate of approximation is possible in general in the spaces employed.

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The effect of interpolating the coefficients in nonlinear parabolic Galerkin procedures

1975-01-01

Jim Douglas , Todd Dupont

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A Galerkin method for a nonlinear Dirichlet problem

1975-01-01

Jim Douglas , Todd Dupont

A Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nabla dot … A Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nabla dot left-parenthesis a left-parenthesis x comma u right-parenthesis nabla u right-parenthesis equals f"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mo>⋅</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\nabla \cdot (a(x,u)\nabla u) = f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The asymptotic error estimates are of the same form as in the linear case. Newton’s method can be used to solve the nonlinear algebraic equations.

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Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces

1974-04-01

Jim Douglas

Some superconvergence results for an H1-Galerkin procedure for the heat equation

1974-01-01

Jim Douglas , Todd Dupont , Mary F. Wheeler

Superconvergence estimates at the knots

1974-01-01

Jim Douglas , Todd Dupont

Collocation Methods for Parabolic Equations in a Single Space Variable

1974-01-01

Jim Douglas , Todd Dupont

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A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems

1974-01-01

Jim Douglas , Todd Dupont , Mary F. Wheeler

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Superconvergence for galerkin methods for the two point boundary problem via local projections

1973-06-01

Jim Douglas , Todd Dupont

Galerkin methods for parabolic equations with nonlinear boundary conditions

1973-06-01

Jim Douglas , Todd Dupont

A finite element collocation method for quasilinear parabolic equations

1973-01-01

Jim Douglas , Todd Dupont

Let the parabolic problem <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c left-parenthesis x comma t comma u right-parenthesis u Subscript t Baseline equals a left-parenthesis x comma t comma u right-parenthesis u … Let the parabolic problem <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c left-parenthesis x comma t comma u right-parenthesis u Subscript t Baseline equals a left-parenthesis x comma t comma u right-parenthesis u Subscript x x Baseline plus b left-parenthesis x comma t comma u comma u Subscript x Baseline right-parenthesis comma 0 greater-than x greater-than 1 comma 0 greater-than t less-than-over-equals upper T comma u left-parenthesis x comma 0 right-parenthesis equals f left-parenthesis x right-parenthesis comma u left-parenthesis 0 comma t right-parenthesis equals g 0 left-parenthesis t right-parenthesis comma u left-parenthesis 1 comma t right-parenthesis equals g 1 left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>x</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>t</mml:mi> <mml:mo>≦</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 > x > 1,0 > t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, be solved approximately by the continuous-time collocation process based on having the differential equation satisfied at Gaussian points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi Subscript i comma 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\xi _{i,1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi Subscript i comma 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\xi _{i,2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in subintervals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript i minus 1 Baseline comma x Subscript i Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">({x_{i - 1}},{x_i})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U colon left-bracket 0 comma upper T right-bracket right-arrow script upper H 3"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>:</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">U:[0,T] \to {\mathcal {H}_3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the class of Hermite piecewise-cubic polynomial functions with knots <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 equals x 0 greater-than x 1 greater-than midline-horizontal-ellipsis greater-than x Subscript n Baseline equals 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>></mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0 = {x_0} > {x_1} > \cdots > {x_n} = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u minus upper U equals upper O left-parenthesis h Superscript 4 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mi>U</mml:mi> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u - U = O({h^4})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> uniformly in <italic>x</italic> and <italic>t</italic>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h equals max left-parenthesis x Subscript i Baseline minus x Subscript i minus 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h = \max ({x_i} - {x_{i - 1}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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A Finite Element Collocation Method for Quasilinear Parabolic Equations

1973-01-01

Jim Douglas , Todd Dupont

Let the parabolic problem $c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 < x < 1,0 < t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t)$, be solved approximately by the continuous-time collocation … Let the parabolic problem $c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 < x < 1,0 < t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t)$, be solved approximately by the continuous-time collocation process based on having the differential equation satisfied at Gaussian points ${\xi _{i,1}}$ and ${\xi _{i,2}}$ in subintervals $({x_{i - 1}},{x_i})$ for a function $U:[0,T] \to {\mathcal {H}_3}$, the class of Hermite piecewise-cubic polynomial functions with knots $0 = {x_0} < {x_1} < \cdots < {x_n} = 1$. It is shown that $u - U = O({h^4})$ uniformly in x and t, where $h = \max ({x_i} - {x_{i - 1}})$.

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Numerical Solutions of the One-Dimensional Primitive Equations Using Galerkin Approximations With localized Basis Functions

1972-10-01

Hsuan-Heng Wang , Paul Halpern , Jim Douglas +1 more

The Galerkin method is applied to a pair of linear and then nonlinear primitive (wave) equations. This results in a system of ordinary differential equations. Procedures are included for generating … The Galerkin method is applied to a pair of linear and then nonlinear primitive (wave) equations. This results in a system of ordinary differential equations. Procedures are included for generating the coefficient matrices of the system of ordinary differential equations when piecewise Hermite cubic functions are used as basis functions. It is demonstrated that this system can be efficiently solved by an implicit method. Numerical examples show that integration using the Galerkin method is more efficient than the corresponding finite-difference method with central differences in space.

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A SUPERCONVERGENCE RESULT FOR THE APPROXIMATE SOLUTION OF THE HEAT EQUATION BY A COLLOCATION METHOD

1972-01-01

Jim Douglas

Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form

1971-01-01

Jim Douglas , Todd Dupont , James Serrin

Galerkin Methods for Parabolic Equations

1970-12-01

Jim Douglas , Todd Dupont

Galerkin-type methods, both continuous and discrete in time, are considered for approximating solutions of linear and nonlinear parabolic problems. Bounds reducing the estimation of the error to questions in approximation … Galerkin-type methods, both continuous and discrete in time, are considered for approximating solutions of linear and nonlinear parabolic problems. Bounds reducing the estimation of the error to questions in approximation theory are derived for the several methods studied. These methods include procedures that lead to linear algebraic equations even for strongly nonlinear problems. A number of computational questions related to these procedures are also discussed.

The Cauchy Problem for the Heat Equation

1967-09-01

John R. Cannon , Jim Douglas

Part I is devoted to the consideration of a Cauchy-like problem for the heat equation. Let $u(x,t)$ satisfy the heat equation in $D = \{ (x,t):0 < x < s(t),0 … Part I is devoted to the consideration of a Cauchy-like problem for the heat equation. Let $u(x,t)$ satisfy the heat equation in $D = \{ (x,t):0 < x < s(t),0 < t \leqq T\} $ and let $u(x,0) = \varphi (x),u(s(t),t) = f(t)$, and $u_x (s(t),t) = g(t)$. An a priori estimate of $| u |$ in terms of the data f, g, $\varphi $ and $\max _D | {u_x } |$ is derived and applied to an error analysis of a numerical procedure for obtaining an approximate solution of the problem. The results of Part I are applied in Part II to an inverse Stef an problem, which consists of determining a heat influx necessary to give a desired free boundary. An a posteriori estimate of the error in the position of the free boundary is determined without the assumption of the existence of a solution of the stated problem, and an a priori bound is derived in the case a solution exists. In Part III the technique of Part I is modified to treat the standard Cauchy problem in which $u(x,0)$ is not specified.

The stability of the boundary in a Stefan problem

1967-01-01

John R. Cannon , Jim Douglas

Multistage Alternating Direction Methods

1966-12-01

Jim Douglas , Arthur Garder , Carl Pearcy

Previous article Next article Multistage Alternating Direction MethodsJim Douglas, Jr., A. O. Garder, and Carl PearcyJim Douglas, Jr., A. O. Garder, and Carl Pearcyhttps://doi.org/10.1137/0703048PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] … Previous article Next article Multistage Alternating Direction MethodsJim Douglas, Jr., A. O. Garder, and Carl PearcyJim Douglas, Jr., A. O. Garder, and Carl Pearcyhttps://doi.org/10.1137/0703048PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Jim Douglas, Jr., Alternating direction methods for three space variables, Numer. Math., 4 (1962), 41–63 10.1007/BF01386295 MR0136083 0104.35001 CrossrefGoogle Scholar[2] Jim Douglas, Jr. and , Carl M. Pearcy, On convergence of alternating direction procedures in the presence of singular operators, Numer. Math., 5 (1963), 175–184 10.1007/BF01385888 MR0154436 0115.34701 CrossrefGoogle Scholar[3] Jim Douglas, Jr. and , H. H. Rachford, Jr., On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc., 82 (1956), 421–439 MR0084194 0070.35401 CrossrefGoogle Scholar[4] D. W. Peaceman and , H. H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), 28–41 10.1137/0103003 MR0071874 0067.35801 LinkISIGoogle Scholar[5] Carl Pearcy, On convergence of alternating direction procedures, Numer. Math., 4 (1962), 172–176 10.1007/BF01386310 MR0145677 0112.34802 CrossrefGoogle Scholar[6] E. L. Wachspress and , G. J. Habetler, An alternating-direction-implicit iteration technique, J. Soc. Indust. Appl. Math., 8 (1960), 403–424 10.1137/0108027 MR0114308 0158.33901 LinkISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Fractional Steps methods for transient problems on commodity computer architecturesPhysics of the Earth and Planetary Interiors, Vol. 171, No. 1-4 Cross Ref Elliptic Equations Cross Ref Computational methods of linear algebraJournal of Soviet Mathematics, Vol. 15, No. 5 Cross Ref In situ measurement of thermal conductivity in the presence of transverse anisotropy20 September 2012 | Journal of Geophysical Research, Vol. 73, No. 16 Cross Ref Volume 3, Issue 4| 1966SIAM Journal on Numerical Analysis History Submitted:11 February 1966Published online:14 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0703048Article page range:pp. 570-581ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

On convergence of alternating direction procedures in the presence of singular operators

1963-12-01

Jim Douglas , Carl Pearcy

Determination of the diffusivity of an isotropic medium

1963-12-01

John R. Cannon , Jim Douglas , B. Frank Jones

On Predictor-Corrector Methods for Nonlinear Parabolic Differential Equations

1963-03-01

Jim Douglas , B. Frank Jones

Previous article On Predictor-Corrector Methods for Nonlinear Parabolic Differential EquationsJim Douglas, Jr. and B. F. Jones, Jr.Jim Douglas, Jr. and B. F. Jones, Jr.https://doi.org/10.1137/0111015PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail … Previous article On Predictor-Corrector Methods for Nonlinear Parabolic Differential EquationsJim Douglas, Jr. and B. F. Jones, Jr.Jim Douglas, Jr. and B. F. Jones, Jr.https://doi.org/10.1137/0111015PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Jim Douglas, Jr., The application of stability analysis in the numerical solution of quasi-linear parabolic differential equations, Trans. Amer. Math. Soc., 89 (1958), 484–518 MR0131673 0084.34702 CrossrefGoogle Scholar[2] Jim Douglas, Jr., Alternating direction methods for three space variables, Numer. Math., 4 (1962), 41–63 10.1007/BF01386295 MR0136083 0104.35001 CrossrefGoogle Scholar[3] Jim Douglas, Jr., A survey of numerical methods for parabolic differential equationsAdvances in Computers, Vol. 2, Academic Press, New York, 1961, 1–54 MR0142211 0133.38503 CrossrefGoogle Scholar[4] Milton Lees, Approximate solutions of parabolic equations, J. Soc. Indust. Appl. Math., 7 (1959), 167–183 10.1137/0107015 MR0110212 0086.32801 LinkISIGoogle Scholar[5] Milton Lees, A priori estimates for the solutions of difference approximations to parabolic partial differential equations, Duke Math. J., 27 (1960), 297–311 10.1215/S0012-7094-60-02727-7 MR0121998 0092.32803 CrossrefISIGoogle Scholar Previous article FiguresRelatedReferencesCited ByDetails A Variable Mesh Finite Difference Method for Solving a Class of Parabolic Differential Equations in One Space VariableSIAM Journal on Numerical Analysis, Vol. 15, No. 4 | 14 July 2006AbstractPDF (1987 KB)A Class of External Flows Exhibiting Self-Similarity in Three-Dimensional Boundary LayersSIAM Journal on Applied Mathematics, Vol. 27, No. 2 | 12 July 2006AbstractPDF (798 KB)Some Computation-Steeples in Fluid MechanicsSIAM Review, Vol. 15, No. 2 | 2 August 2006AbstractPDF (2556 KB) Volume 11, Issue 1| 1963Journal of the Society for Industrial and Applied Mathematics1-204 History Submitted:06 December 1961Published online:13 July 2006 InformationCopyright © 1963 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0111015Article page range:pp. 195-204ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

Two high-order correct difference analogues for the equation of multidimensional heat flow

1963-01-01

Jim Douglas , James E. Gunn

1. Introduction. Several high-order accuracy difference equations for the heat equation in one space variable [1] have been proposed, but most do not extend to several space variables with any … 1. Introduction. Several high-order accuracy difference equations for the heat equation in one space variable [1] have been proposed, but most do not extend to several space variables with any ease, if at all. Two three-level difference equations are discussed here, each of which is fourth-order correct in space and second in time. One is stable and convergent in 22 for as many as four space variables but is limited essentially to the heat equation itself and in the four space variables case to bounded r = At( Ax) -2. The other is stable and convergent in 22 for three space variables and is adapted to extension to more complicated differential equations. Alternating direction techniques based on the two three-level formulas are developed. These methods retain the accuracy of the original procedures and require much less arithmetic to complete a problem. Only the results will be given here; their analyses will be presented in another paper [3] as examples of a general approach to alternating direction methods.

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Two High-Order Correct Difference Analogues for the Equation of Multidimensional Heat Flow

1963-01-01

Jim Douglas , James E. Gunn

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Common Coauthors

Coauthor	Papers Together
Todd Dupont	22
John R. Cannon	4
Mary F. Wheeler	4
Seongjai Kim	3
Douglas N. Arnold	3
Felipe Pereira	3
James E. Gunn	3
Thomas M. Gallie	3
B. Frank Jones	3
H.H. Rachford	2
Hyeona Lim	2
Lars B. Wahlbin	2
Vidar Thomée	2
Jean E. Roberts	2
Carl Pearcy	2
Eduardo Abreu	1
Guangyu Li	1
Chieh‐Sen Huang	1
Daoqi Yang	1
Thomas F. Russell	1
César Almeida	1
James Serrin	1
Fabio Milner	1
Richard E. Ewing	1
Mary F. Wheeler	1
H. H. Rachford	1
Chaitan P. Gupta	1
Jeffrey L. Hensley	1
D. Marchesin	1
Arthur Garder	1
Juan E. Santos	1
Luis Carlos Roman	1
Paul Halpern	1
Dongwoo Sheen	1
Li-Ming Yeh	1
Frederico Furtado	1
Hsuan-Heng Wang	1
T. M. Gallie	1

Commonly Cited References

The Numerical Solution of Parabolic and Elliptic Differential Equations

1955-03-01

D.W. Peaceman , H.H. Rachford

Previous article Next article The Numerical Solution of Parabolic and Elliptic Differential EquationsD. W. Peaceman and H. H. Rachford, Jr.D. W. Peaceman and H. H. Rachford, Jr.https://doi.org/10.1137/0103003PDFPDF PLUSBibTexSections ToolsAdd to … Previous article Next article The Numerical Solution of Parabolic and Elliptic Differential EquationsD. W. Peaceman and H. H. Rachford, Jr.D. W. Peaceman and H. H. Rachford, Jr.https://doi.org/10.1137/0103003PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. H., Bruce, , D. W., Peaceman, , H. H., Rachford and , J. D. Rice, Calculation of unsteady-state gas flow through porous media, Trans. Amer. Inst. Mining and Met. Engrs., 198 (1953), 79– ISIGoogle Scholar[2] H. S. Carslaw and , J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947viii+386 MR0022294 Google Scholar[3] Stanley P. Frankel, Convergence rates of iterative treatments of partial differential equations, Math. Tables and Other Aids to Computation, 4 (1950), 65–75 MR0046149 CrossrefGoogle Scholar[4] George G. O'Brien, , Morton A. Hyman and , Sidney Kaplan, A study of the numerical solution of partial differential equations, J. Math. Physics, 29 (1951), 223–251 MR0040805 0042.13204 CrossrefISIGoogle Scholar[5] Jim Douglas, Jr., On the numerical integration of $\partial\sp 2u/\partial x\sp 2+\partial\sp 2u/\partial y\sp 2=\partial u/\partial t$ by implicit methods, J. Soc. Indust. Appl. Math., 3 (1955), 42–65 10.1137/0103004 MR0071875 0067.35802 LinkISIGoogle Scholar[6] Jim Douglas, Jr. and , T. M. Gallie, Jr., Variable time steps in the solution of the heat flow equation by a difference equation, Proc. Amer. Math. Soc., 6 (1955), 787–793 MR0078754 0066.10502 CrossrefGoogle Scholar[7] J. Douglas, Jr. and , D. W. Peaceman, Numerical solution of two-dimensional heat flow problems, to be presented at the May, 1955 meeting of The American Institute of Chemical Engineers at Houston, Texas Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A General Alternating-Direction Implicit Framework with Gaussian Process Regression Parameter Prediction for Large Sparse Linear SystemsKai Jiang, Xuehong Su, and Juan ZhangSIAM Journal on Scientific Computing, Vol. 44, No. 4 | 7 July 2022AbstractPDF (1383 KB)Convergence Analysis of the Nonoverlapping Robin--Robin Method for Nonlinear Elliptic EquationsEmil Engström and Eskil HansenSIAM Journal on Numerical Analysis, Vol. 60, No. 2 | 21 March 2022AbstractPDF (565 KB)Robust Alternating Direction Implicit Solver in Quantized Tensor Formats for a Three-Dimensional Elliptic PDEM. 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A Study of the Numerical Solution of Partial Differential Equations

1950-04-01

George G. O’Brien , Morton A. Hyman , Sidney Kaplan

On the Relation Between Stability and Convergence in the Numerical Solution of Linear Parabolic and Hyperbolic Differential Equations

1956-03-01

Jim Douglas

On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit Methods

1955-03-01

Jim Douglas

Previous article On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit MethodsJim Douglas, Jr.Jim Douglas, Jr.https://doi.org/10.1137/0103004PDFBibTexSections ToolsAdd … Previous article On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit MethodsJim Douglas, Jr.Jim Douglas, Jr.https://doi.org/10.1137/0103004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. H. Bruce, , D. W. Peaceman, , H. H. Rachford and , J. D. Rice, Calculations of unsteady-state gas flow through porous media, Journal of Petroleum technology, 198 (1953), 79–91 CrossrefGoogle Scholar[2] H. Lewy, , R. Courant and , K. Friedrichs, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32–74 10.1007/BF01448839 MR1512478 CrossrefGoogle Scholar[3] J. Crank and , P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. Cambridge Philos. Soc., 43 (1947), 50–67 MR0019410 CrossrefISIGoogle Scholar[4] S. Hartree and , J. R. Womersley, A method for the numerical or mechanical solution of certain types of partial differential equations, Proc. Royal Soc. London, Series A, 161 (1937), 353–366 0017.08003 CrossrefGoogle Scholar[5] F. B. Hildebrand, On the convergence of numerical solutions of the heat-flow equation, J. Math. Physics, 31 (1952), 35–41 MR0048171 0048.07602 CrossrefISIGoogle Scholar[6] Fritz John, On integration of parabolic equations by difference methods. I. Linear and quasi-linear equations for the infinite interval, Comm. Pure Appl. Math., 5 (1952), 155–211 MR0047885 0047.33703 CrossrefISIGoogle Scholar[7] M. L. Juncosa and , D. M. Young, On the order of convergence of solutions of a difference equation to a solution of the diffusion equation, J. Soc. Indust. Appl. Math., 1 (1953), 111–135 10.1137/0101007 MR0060907 0053.46303 LinkGoogle Scholar[8] Pentti Laasonen, Über eine Methode zur Lösung der Wärmeleitungsgleichung, Acta Math., 81 (1949), 309–317 MR0032094 0040.35802 CrossrefISIGoogle Scholar[9] Werner Leutert, On the convergence of approximate solutions of the heat equation to the exact solution, Proc. Amer. Math. Soc., 2 (1951), 433–439 MR0043577 0043.12602 CrossrefISIGoogle Scholar[10] George G. O'Brien, , Morton A. Hyman and , Sidney Kaplan, A study of the numerical solution of partial differential equations, J. Math. Physics, 29 (1951), 223–251 MR0040805 0042.13204 CrossrefISIGoogle Scholar[11] D. W. Peaceman and , H. H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), 28–41 10.1137/0103003 MR0071874 0067.35801 LinkISIGoogle Scholar[12] Erich Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann., 102 (1930), 650–670 10.1007/BF01782368 MR1512599 CrossrefGoogle Scholar Previous article FiguresRelatedReferencesCited byDetails Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field ModelsMatthew Causley, Hana Cho, and Andrew Christlieb12 October 2017 | SIAM Journal on Scientific Computing, Vol. 39, No. 5AbstractPDF (2175 KB)Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive Convolution2 June 2016 | SIAM Journal on Numerical Analysis, Vol. 54, No. 3AbstractPDF (1298 KB)Fourth Order Accurate Scheme for the Space Fractional Diffusion Equations12 June 2014 | SIAM Journal on Numerical Analysis, Vol. 52, No. 3AbstractPDF (453 KB)An $h$-Adaptive Operator Splitting Method for Two-Phase Flow in 3D Heterogeneous Porous Media29 January 2013 | SIAM Journal on Scientific Computing, Vol. 35, No. 1AbstractPDF (1869 KB)Difference Graphs of Block ADI Method26 July 2006 | SIAM Journal on Numerical Analysis, Vol. 38, No. 3AbstractPDF (162 KB)Domain Decomposition Operator Splittings for the Solution of Parabolic Equations25 July 2006 | SIAM Journal on Scientific Computing, Vol. 19, No. 3AbstractPDF (406 KB)Alternating Direction Implicit Methods for Parabolic Equations with a Mixed Derivative16 May 2012 | SIAM Journal on Scientific and Statistical Computing, Vol. 1, No. 1AbstractPDF (2857 KB)Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 3AbstractPDF (2718 KB)Rounding Errors in Alternating Direction Methods for Parabolic Problems3 August 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 2AbstractPDF (1267 KB)Alternating Direction Schemes for the Heat Equation in a General Domain14 July 2006 | Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, Vol. 2, No. 3AbstractPDF (1121 KB)On Incomplete Iteration for Implicit Parabolic Difference Equations10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 3AbstractPDF (466 KB)An Alternating-Direction-Implicit Iteration Technique10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 2AbstractPDF (1508 KB)On the Relation Between Stability and Convergence in the Numerical Solution of Linear Parabolic and Hyperbolic Differential Equations10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 4, No. 1AbstractPDF (1316 KB)The Numerical Solution of Parabolic and Elliptic Differential EquationsD. W. Peaceman and H. H. Rachford, Jr.10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 3, No. 1AbstractPDF (1199 KB) Volume 3, Issue 1| 1955Journal of the Society for Industrial and Applied Mathematics History Submitted:18 October 1954Published online:10 July 2006 InformationCopyright © 1955 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0103004Article page range:pp. 42-65ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

On the numerical integration of quasilinear parabolic differential equations

1956-03-01

Jim Douglas

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A Priori $L_2 $ Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations

1973-09-01

Mary F. Wheeler

$L_2 $ error estimates for the continuous time and several discrete time Galerkin approximations to solutions of some second order nonlinear parabolic partial differential equations are derived. Both Neumann and … $L_2 $ error estimates for the continuous time and several discrete time Galerkin approximations to solutions of some second order nonlinear parabolic partial differential equations are derived. Both Neumann and Dirichlet boundary conditions are considered. The estimates obtained are the best possible in an $L_2 $ sense. These error estimates are derived by relating the error for the nonlinear parabolic problem to known $L_2 $ error estimates for a linear elliptic problem. With additional restrictions on basis functions and region $L_\infty $ error estimates are derived. Possible extensions to other discrete time Galerkin procedures and to higher order parabolic equations and systems of parabolic equations are suggested.

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On the numerical solution of heat conduction problems in two and three space variables

1956-01-01

Jim Douglas , H.H. Rachford

f(iAx f(iAx

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Numerical solution of differential equations

2011-03-23

W. E. Milne

A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type

1947-01-01

J Crank , P. Nicolson

This paper is concerned with methods of evaluating numerical solutions of the non-linear partial differential equation where subject to the boundary conditions A, k, q are known constants. Equation (1) … This paper is concerned with methods of evaluating numerical solutions of the non-linear partial differential equation where subject to the boundary conditions A, k, q are known constants. Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2).

Variable Time Steps in the Solution of the Heat Flow Equation by a Difference Equation

1955-10-01

Jim Douglas , T. M. Gallie

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On the Numerical Solution of Heat Conduction Problems in Two and Three Space Variables

1956-07-01

Jim Douglas , H. H. Rachford

f(iAx f(iAx

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On the convergence of approximate solutions of the heat equation to the exact solution

1951-06-01

Werner Leutert

Introduction. In many cases it is practically impossible to solve an initial value problem for a partial differential equation exactly although it can be proved that the exact solution does … Introduction. In many cases it is practically impossible to solve an initial value problem for a partial differential equation exactly although it can be proved that the exact solution does exist and is uniquely determined. Therefore the partial differential equation is very often replaced by a difference equation which is easier to solve and which furnishes an approximation to the solution of the original problem. Three questions arise immediately. (a) If x and t are the independent variables, does the mesh ratio r (r=At/(Ax)2 in the parabolic case) have any influence on the convergence or stability of the approximate solution? (b) Which one of the difference equations leading to the differential equation will furnish a good approximation? (c) How do the initial values of the problem for the difference equation have to be chosen in order to furnish a good approximation? The author believes that the first question has been overemphasized while the importance of the second and third has not been fully realized. This is primarily due to the fact that, in the paper by O'Brien, Hyman, and Kaplan []1 where von Neumann's test of stability is introduced into the literature, it is erroneously stated that a positive answer to von Neumann's test is necessary and sufficient for convergence. As an example it is pointed out in their paper that the numerical solution of Richardson [2] for the problem of finding the temperature in a slab with faces at temperature zero does not converge to the exact solution because von Neumann's test shows instability for all r> 0.2

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Numerical Solution of Differential Equations.

1953-01-01

K. D. Tocher , W. E. Milne

On the Order of Convergence of Solutions of a Difference Equation to a Solution of the Diffusion Equation

1953-12-01

M. L. Juncosa , Dean M. Young

Previous article Next article On the Order of Convergence of Solutions of a Difference Equation to a Solution of the Diffusion EquationM. L. Juncosa and D. M. YoungM. L. Juncosa … Previous article Next article On the Order of Convergence of Solutions of a Difference Equation to a Solution of the Diffusion EquationM. L. Juncosa and D. M. YoungM. L. Juncosa and D. M. Younghttps://doi.org/10.1137/0101007PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] W. E. Byerly, Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, Ginn and Co., Boston, 1893 Google Scholar[2] H. S. Carslaw, The Mathematical Theory of the Conduction of Heat in Solids, MacMillan, London, 1921 Google Scholar[3] M. L. Cartwright, On the Relation Between Different Types of Abel Summation, Proc. Lond. Math. Soc. (2), 31 (1930), 81–96 CrossrefGoogle Scholar[4] Ruel V. Churchill, Fourier Series and Boundary Value Problems, McGraw-Hill Book Company, Inc., New York and London, 1941ix+206 MR0003251 (2,189d) Google Scholar[5] J. Crank and , P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. Cambridge Philos. Soc., 43 (1947), 50–67 MR0019410 (8,409b) CrossrefISIGoogle Scholar[6] Leopold Fejér, Untersuchungen über Fouriersche Reihen, Math. Ann., 58 (1903), 51–69 MR1511228 CrossrefGoogle Scholar[7] W. Fulks, On the solutions of the heat equation, Proc. Amer. Math. Soc., 2 (1951), 973–979 MR0046542 (13,750f) 0045.04803 CrossrefISIGoogle Scholar[8] M. Gévrey, Sur les équations aux dérivées du type parabolique, Jour. d. Math., ser. 6, 6 (1913), 305–471 Google Scholar[9] E. Goursat, Cours d'analyse mathématique, Vol. 3, Paris, 1927 Google Scholar[10] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949xvi+396 MR0030620 (11,25a) 0032.05801 Google Scholar[11] G. H. Hardy and , W. W. Rogosinski, Fourier Series, Cambridge Tracts in Mathematics and Mathematical Physics, no. 38, Cambridge University Press, 1944, 100– MR0010206 (5,261d) 0060.18208 Google Scholar[12] Philip Hartman and , Aurel Wintner, On the solutions of the equation of heat conduction, Amer. J. Math., 72 (1950), 367–395 MR0036412 (12,104a) 0038.25801 CrossrefISIGoogle Scholar[13] F. B. Hildebrand, On the convergence of numerical solutions of the heat-flow equation, J. Math. 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Monthly, 44 (1937), 155–160 MR1523881 0016.29901 CrossrefGoogle Scholar[24] J. L. Walsh and , David Young, On the degree of convergence of solutions of difference equations to the solution of the Dirichlet problem, J. Math. Physics, 33 (1954), 80–93 MR0060908 (15,746e) 0055.08901 CrossrefISIGoogle Scholar[25] Wolfgang Wasow, On the truncation error in the solution of Laplace's equation by finite differences, J. Research Nat. Bur. Standards, 48 (1952), 345–348 MR0048923 (14,93g) CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Rates of Convergence of Approximate Solutions of Parabolic Initial-Boundary Value ProblemsC. S. Caldwell14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 11, No. 6AbstractPDF (1209 KB)Stability Theory for Partial Difference OperatorsVidar Thomée18 July 2006 | SIAM Review, Vol. 11, No. 2AbstractPDF (3792 KB)The Rate of Convergence of Some Difference SchemesG. W. Hedstrom3 August 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 2AbstractPDF (3260 KB)On the Relation Between Stability and Convergence in the Numerical Solution of Linear Parabolic and Hyperbolic Differential EquationsJim Douglas, Jr.10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 4, No. 1AbstractPDF (1316 KB)On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit MethodsJim Douglas, Jr.10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 3, No. 1AbstractPDF (1365 KB) Volume 1, Issue 2| 1953Journal of the Society for Industrial and Applied Mathematics History Submitted:05 August 1953Published online:28 July 2006 InformationCopyright © 1953 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0101007Article page range:pp. 111-135ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

The Solution of the Diffusion Equation by a High Order Correct Difference Equation

1956-04-01

Jim Douglas

A Survey of Numerical Methods for Parabolic Differential Equations

1961-01-01

Jim Douglas

On the Convergence of Numerical Solutions of the Heat‐Flow Equation

1952-04-01

F. B. Hildebrand

Rayleigh‐Ritz‐Galerkin methods for dirichlet's problem using subspaces without boundary conditions

1970-07-01

James H. Bramble , A. H. Schatz

Variable time steps in the solution of the heat flow equation by a difference equation

1955-10-01

Jim Douglas , Thomas M. Gallie

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An Optimal $L_\infty $ Error Estimate for Galerkin Approximations to Solutions of Two-Point Boundary Value Problems

1973-10-01

Mary F. Wheeler

An a priori $L_\infty $ error estimate is established for continuous piecewise polynomial Galerkin approximations to solutions of linear two-point boundary value problems. This estimate is the best possible in … An a priori $L_\infty $ error estimate is established for continuous piecewise polynomial Galerkin approximations to solutions of linear two-point boundary value problems. This estimate is the best possible in that the order of convergence is optimal and the norm on the solutions cannot be weakened.

A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems

1974-01-01

Jim Douglas , Todd Dupont , Mary F. Wheeler

avec les conditions générales d'utilisation (http://www.numdam.org/legal. avec les conditions générales d'utilisation (http://www.numdam.org/legal.

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An observation concerning Ritz-Galerkin methods with indefinite bilinear forms

1974-01-01

A. H. Schatz

Existence, uniqueness and error estimates for Ritz-Galerkin methods are discussed in the case where the associated bilinear form satisfies a Gårding type inequality, i.e., it is indefinite in a certain … Existence, uniqueness and error estimates for Ritz-Galerkin methods are discussed in the case where the associated bilinear form satisfies a Gårding type inequality, i.e., it is indefinite in a certain way. An application to the finite element method is given.

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Two-Level Discrete-Time Galerkin Approximations for Second Order Nonlinear Parabolic Partial Differential Equations

1973-12-01

H. H. Rachford

Galerkin methods for nonlinear parabolic problems are studied. For suitable space domains in $\mathbb{R}^3 $, two-level discrete-time methods are proposed which are shown to yield optimal order of approximation for … Galerkin methods for nonlinear parabolic problems are studied. For suitable space domains in $\mathbb{R}^3 $, two-level discrete-time methods are proposed which are shown to yield optimal order of approximation for a more general class of problem than has been analyzed heretofore. An induction argument is used to treat nonlinear capacity terms. One method studied yields second order convergence in time while requiring the solution of a linear algebraic system only once per time step.

Collocation Methods for Parabolic Equations in a Single Space Variable

1974-01-01

Jim Douglas , Todd Dupont

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The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations

1972-01-01

Abdul Kadir Aziz

Lectures on Elliptic Boundary Value Problems.

1966-06-01

A. Friedman , Shmuel Agmon

An Observation Concerning Ritz-Galerkin Methods with Indefinite Bilinear Forms

1974-10-01

A. H. Schatz

Existence, uniqueness and error estimates for Ritz-Galerkin methods are o discussed in the case where the associated bilinear form satisfies a Carding type inequality, i.e., it is indefinite in a … Existence, uniqueness and error estimates for Ritz-Galerkin methods are o discussed in the case where the associated bilinear form satisfies a Carding type inequality, i.e., it is indefinite in a certain way.An application to the finite element method is given.In this note, we would like to discuss existence, uniqueness and estimates over the whole domain for some Ritz-Galerkin methods where the bilinear form satisfies o a Garding type inequality, i.e., it is indefinite in a special way.We shall first illustrate the problem by an example.For simplicity, let £2 be a simply connected convex region in the plane with a polygonal boundary d£2 and consider the Dirichlet problem 2 2 (1) Lu= £ -D.(a..(xyDu) + ]y b.(x)D.u+ c(xyu=f in £2, u = 0 on 3£2, i,j=l ' ,; ; ¡=i ' 'where L is uniformly elliptic in £2; for simplicity, we assume that the coefficients belong to Cl(£l).Let us suppose that for each /G Z,2(£2), the problem (1) has a unique solution u.It is then well known that u G W2(iï.) n W2(£2).Suppose that we wish to approximate u using the finite element method.For this, we subdivide £2 into triangles with largest side A and smallest angle a > aQ > 0 and define a .0 1 finite-dimensional subspace S C W2 to be, for example, the set of piecewise linear functions on this triangulation which vanish on 9£2.We then seek to determine an approximate solution uh G Sh from the Ritz-Galerkin equations B(uh, tp) = B(u, xp) = j j fipdx, for all xp G Sh, (2) « B(u, xp) = ¡j I ¿ a(Du)(Dip) + £ b(Du)ip + cuip) dx. a \ij=i ' ' í=i J Let us note the inequality (3) \B(u, ip)\ < CIImII, MIj, for all u, xp G ^(£2).

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Approximate Solutions of Parabolic Equations

1959-06-01

Milton Lees

On integration of parabolic equations by difference methods: I. Linear and quasi‐linear equations for the infinite interval

1952-05-01

Fritz John

Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems

1975-01-01

Jim Douglas , Todd Dupont , Lars B. Wahlbin

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Piecewise Hermite interpolation in one and two variables with applications to partial differential equations

1968-03-01

G. Birkhoff , Martin H. Schultz , R. S. Varga

Galerkin Methods for Parabolic Equations

1970-12-01

Jim Douglas , Todd Dupont

Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form

1971-01-01

Jim Douglas , Todd Dupont , James Serrin

A note on the alternating direction implicit method for the numerical solution of heat flow problems

1957-04-01

Jim Douglas

1. Introduction. In companion papers [1; 6] recently Peaceman, Rachford, and the author introduced a finite difference technique called therein the alternating direction implicit method for approximating the solution of … 1. Introduction. In companion papers [1; 6] recently Peaceman, Rachford, and the author introduced a finite difference technique called therein the alternating direction implicit method for approximating the solution of transient and permanent heat flow problems in two space variables. The validity of the method was established only in the case of a rectangular domain. Since then the procedure has been tested successfully on several more complex examples [4] without proof. The purpose of this short note is to prove in the case of non rectangular domains that the solution of the alternating direction method for the parabolic problem converges to the solution of the differential equation as the increments of the independent variables diminish in a proper manner, that the iterative adaptation for the elliptic problem converges to the solution of the Laplace difference equation, and to give an efficient choice of the parameter sequence involved in this iteration. 2. Parabolic problem. Let D be an open, connected set in the plane bounded by a curve C made up of closed polygons with sides parallel to the coordinate axes. Assume, moreover, that there exists a sequence

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Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces

1974-04-01

Jim Douglas

On the convergence of a solution of a difference equation to a solution of the equation of diffusion

1954-02-01

M. L. Juncosa , Dean M. Young

Furthermore, (1) satisfies the boundary conditions u( + 0, t) =u(1 -0, t) =0, t>0 and as a consequence of ?1 and ?3 of Appendix II of [3] it also … Furthermore, (1) satisfies the boundary conditions u( + 0, t) =u(1 -0, t) =0, t>0 and as a consequence of ?1 and ?3 of Appendix II of [3] it also satisfies the initial condition u(x, +0) =f(x) at every point of continuity off(x) in 0 <x< 1 as well as the condition u(x, +0) =(1/2)[f(x+0)+f(x-0)] at every point x where f(x) possesses these one-sided limits. Moreover, from a slight modification of Hardy and Rogosinski [4, p. 66] on Abel summability of Fourier series and from Theorems 270 (due to M. L. Cartwright [1]) and 273 in Appendix V of [3] it follows that as t-0O +,

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Methods of Numerical Mathematics

1977-01-01

Bertil Gustafsson , G. I. Marchuk

1 Fundamentals of the Theory of Difference Schemes.- 1.1. Basic Equations and Their Adjoints.- 1.1.1. Norm Estimates of Certain Matrices.- 1.1.2. Computing the Spectral Bounds of a Positive Matrix.- 1.1.3. … 1 Fundamentals of the Theory of Difference Schemes.- 1.1. Basic Equations and Their Adjoints.- 1.1.1. Norm Estimates of Certain Matrices.- 1.1.2. Computing the Spectral Bounds of a Positive Matrix.- 1.1.3. Eigenvalues and Eigenfunctions of the Laplace Operator.- 1.1.4. Eigenvalues and Eigenvectors of the Finite-Difference Analog of the Laplace Operator.- 1.2. Approximation.- 1.3. Countable Stability.- 1.4. The Convergence Theorem.- 2 Methods of Constructing Difference Schemes for Differential Equations.- 2.1. Variational Methods in Mathematical Physics.- 2.1.1. Some Problems of Variational Calculation.- 2.1.1. The Ritz Method.- 2.1.3. The Galerkin Method.- 2.1.4. The Method of Least Squares.- 2.2. The Method of Integral Identities.- 2.2.1. Method of Constructing Difference Equations for Problems with Discontinuous Coefficients on the Basis of an Integral Identity.- 2.2.2. The Variational Form of an Integral Identity.- 2.3. Difference Schemes for Equations with Discontinuous Coefficients Based on Variational Principles.- 2.3.1. Simple Difference Equations for a Diffusion Based on the Ritz Method.- 2.3.2. Constructions of Simple Difference Schemes Based on the Galerkin (Finite Elements) Method.- 2.4. Principles for the Construction of Subspaces for the Solution of One-Dimensional Problems by Variational Methods.- 2.4.1. A General Approach to the Construction of Subspaces of Piecewise-Polynomial Functions.- 2.4.2. Constructing a Basis Using Trigonometric Functions and Applying It in Variational Methods.- 2.5. Variational-Difference Schemes for Two-Dimensional Equations of Elliptic Type.- 2.5.1. The Ritz Method.- 2.5.2. The Galerkin Method.- 2.5.3. Methods for Constructing Subspaces.- 2.6. Variational Methods for Multi-Dimensional Problems.- 2.6.1. Methods of Choosing the Subspaces.- 2.6.2. Coordinate-by-Coordinate Methods for Multi-Dimensional Problems.- 2.7. The Method of Fictive Domains.- 3 Interpolation of Net Functions.- 3.1. Interpolation of Functions of One Variable.- 3.1.1. Interpolation of Functions of One Variable by Cubic Splines.- 3.1.2. Piecewise-Cubic Interpolation with Smoothing.- 3.1.3. Smooth Construction.- 3.1.4. The Convergence of Spline Functions.- 3.2. Interpolation of Functions of Two or More Variables.- 3.3. An r-Smooth Approximation to a Function of Several Variables.- 3.4. Elements of the General Theory of Splines.- 4 Methods for Solving Stationary Problems of Mathematical Physics.- 4.1. General Concepts of Iteration Theory.- 4.2. Some Iterative Methods and Their Optimization.- 4.2.1. The Simplest Iteration Method.- 4.2.2. Convergence and Optimization of Stationary Iterative Methods.- 4.2.3. The Successive Over-Relaxation Method.- 4.2.4. The Chebyshev Iteration Method.- 4.2.5. Comparison of the Convergence Rates of Various Iteration Methods for a System of Finite-Difference Equations.- 4.3. Nonstationary Iteration Methods.- 4.3.1. Convergence Theorems.- 4.3.2. The Method of Minimizing the Residuals.- 4.3.3. The Conjugate Gradient Method.- 4.4. The Splitting-Up Method.- 4.4.1. The Commutative Case.- 4.4.2. The Noncommutative Case.- 4.4.3. Variational and Chebyshevian Optimization of Splitting-Up Methods.- 4.5. Iteration Methods for Systems with Singular Matrices.- 4.5.1. Consistent Systems.- 4.5.2. Inconsistent Systems.- 4.5.3. The Matrix Analog of the Method of Fictive Regions.- 4.6. Iterative Methods for Inaccurate Input Data.- 4.7. Direct Methods for Solving Finite-Difference Systems.- 4.7.1. The Fast Fourier Transform.- 4.7.2. The Cyclic Reduction Method.- 4.7.3. Factorization of Difference Equations.- 5 Methods for Solving Nonstationary Problems.- 5.1. Second-Order Approximation Difference Schemes with Time-Varying Operators.- 5.2. Nonhomogeneous Equations of the Evolution Type.- 5.3. Splitting-Up Methods for Nonstationary Problems.- 5.3.1. The Stabilization Method.- 5.3.2. The Predictor-Corrector Method.- 5.3.3. The Component-by-Component Splitting-Up Method.- 5.3.4. Some General Remarks.- 5.4. Multi-Component Splitting.- 5.4.1. The Stabilization Method.- 5.4.2. The Predictor-Corrector Method.- 5.4.3. The Component-by-Component Splitting-Up Method Based on the Elementary Schemes.- 5.4.4. Splitting-Up of Quasi-Linear Problems.- 5.5. General Approach to Component-by-Component Splitting.- 5.6. Methods of Solving Equations of the Hyperbolic Type.- 5.6.1. The Stabilization Method.- 5.6.2. Reduction of the Wave Equation to an Evolution Problem.- 6 Richardson's Method for Increasing the Accuracy of Approximate Solutions.- 6.1 Ordinary First-Order Differential Equations.- 6.2. General Results.- 6.2.1. The Decomposition Theorem.- 6.2.2. Acceleration of Convergence.- 6.3. Simple Integral Equations.- 6.3.1. The Fredholm Equation of the Second Kind.- 6.3.2. The Volterra Equation of the First Kind.- 6.4. The One-Dimensional Diffusion Equation.- 6.4.1. The Difference Method.- 6.4.2. The Galerkin Method.- 6.5. Nonstationary Problems.- 6.5.1. The Heat Equation.- 6.5.2. The Splitting-Up Method for the Evolutionary Equation.- 6.6. Richardson's Extrapolation for Multi-Dimensional Problems.- 7 Numerical Methods for Some Inverse Problems.- 7.1. Fundamental Definitions and Examples.- 7.2. Solution of the Inverse Evolution Problem with a Constant Operator.- 7.2.1. The Fourier Method.- 7.2.2. Reduction to the Solution of a Direct Equation.- 7.3. Inverse Evolution Problems with Time-Varying Operators.- 7.4. Methods of Perturbation Theory for Inverse Problems.- 7.4.1. Some Problems of the Linear Theory of Measurements.- 7.4.2. Conjugate Functions and the Notion of Value.- 7.4.3. Perturbation Theory for Linear Functionals.- 7.4.4. Numerical Methods for Inverse Problems and Design of Experiment.- 7.5. Perturbation Theory for Complex Nonlinear Models.- 7.5.1. Fundamental and Adjoint Equations.- 7.5.2. The Adjoint Equation in Perturbation Theory.- 7.5.3. Perturbation Theory for Nonstationary Problems.- 7.5.4. Spectral Methods in Perturbation Theory.- 8 Methods of Optimization.- 8.1. Convex Programming.- 8.2. Linear Programming.- 8.3. Quadratic Programming.- 8.4. Numerical Methods in Convex Programming Problems.- 8.5. Dynamic Programming.- 8.6. Pontrjagin's Maximum Principle.- 8.7. Extremal Problems with Constraints and Variational Inequalities.- 8.7.1. Elements of the General Theory.- 8.7.2. Examples of Extremal Problems.- 8.7.3. Numerical Methods in Extremal Problems.- 9 Some Problems of Mathematical Physics.- 9.1. The Poisson Equation.- 9.1.1. The Dirichlet Problem for the One-Dimensional Poisson Equation.- 9.1.2. The One-Dimensional von Neumann Problem.- 9.1.3. The Two-Dimensional Poisson Equation.- 9.1.4. A Problem of Boundary Conditions.- 9.2. The Heat Equation.- 9.2.1. The One-Dimensional Problem of Heat Conduction.- 9.2.2. The Two-Dimensional Problem of Heat Conduction.- 9.3. The Wave Equation.- 9.4. The Equation of Motion.- 9.4.1. The Simplest Equations of Motion.- 9.4.2. The Two-Dimensional Equation of Motion with Variable Coefficients.- 9.4.3. The Multi-Dimensional Equation of Motion.- 9.5. The Neutron Transport Equation.- 9.5.1. The Nonstationary Equation.- 9.5.2. The Transport Equation in Self-Adjoint Form.- 10 A Review of the Methods of Numerical Mathematics.- 10.1. The Theory of Approximation, Stability, and Convergence of Difference Schemes.- 10.2. Numerical Methods for Problems of Mathematical Physics.- 10.3. Conditionally Well-Posed Problems.- 10.4. Numerical Methods in Linear Algebra.- 10.5. Optimization Problems in Numerical Methods.- 10.6. Optimization Methods.- 10.7. Some Trends in Numerical Mathematics.- References.- Index of Notation.

Survey of the stability of linear finite difference equations

1956-05-01

Peter D. Lax , Robert D. Richtmyer

Numerical Solution of Partial Differential Equations.

1966-12-01

G. W. Hedstrom , G. D. Smith

Numerical Solution of Sobolev Partial Differential Equations

1975-06-01

Richard E. Ewing

Finite difference techniques can be applied to the numerical solution of the initial-boundary value problem in S for the semilinear Sobolev or pseudo-parabolic equation \[ \sum\limits_{i = 1}^n {\left[ {\frac{\partial … Finite difference techniques can be applied to the numerical solution of the initial-boundary value problem in S for the semilinear Sobolev or pseudo-parabolic equation \[ \sum\limits_{i = 1}^n {\left[ {\frac{\partial } {{\partial x_i }}\left( {a_i \frac{\partial } {{\partial x_i }}u_t } \right) + \frac{\partial } {{\partial x_i }}\left( {b_i \frac{\partial } {{\partial x_i }}u} \right)} \right]} - q = ru_t , \] where $a_i $, $b_i $, q and r are functions of space and time variables, q is a boundedly differentiable function of u, and S is an open, connected domain in $\mathbb{R}^n $. Under suitable smoothness conditions, the solution of a Crank–Nicolson type of difference equation is shown to converge to u in the discrete $L^2 $-norm with an $O((\Delta x)^2 + (\Delta t)^2 )$ discretization error. The numerical problem is reduced to the inversion of a certain matrix at each time level. For the problem with constant coefficients in a two- or three-dimensional cube, a two-level iteration scheme with a Picard-type outer iteration and an alternating direction inner iteration is presented. For more general operators and more general regions in $\mathbb{R}^n $ for arbitrary n the same two-level scheme with a successive overrelaxation inner iteration is discussed.

On the numerical integration of a parabolic differential equation subject to a moving boundary condition

1955-12-01

Jim Douglas , Thomas M. Gallie

The initial-value problem for the Korteweg-de Vries equation

1975-07-03

Jerry L. Bona , R. Jeffrey Smith

For the Korteweg-de Vries equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> … For the Korteweg-de Vries equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , existence, uniqueness, regularity and continuous dependence results are established for both the pure initial-value problem (posed on -∞< x <∞) and the periodic initial-value problem (posed on 0 ⩽ x ⩽ l with periodic initial data). The results are sharper than those obtained previously in that the solutions provided have the same number of L 2 -derivatives as the initial data and these derivatives depend continuously on time, as elements of L 2 . The same equation with dissipative and forcing terms added is also examined. A by-product of the methods used is an exact relation between solutions of the Korteweg-de Vries equation and solutions of an alternative model equation recently studied by Benjamin, Bona & Mahony (1972). It is proven that in the long-wave limit under which these equations are generally derived, the solutions of the two models posed for the same initial data are the same. In the process of carrying out this programme, new results are obtained for the latter model equation.

Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections

1972-01-01

James H. Bramble , Todd Dupont , Vidar Thomée

Consider Dirichlet’s problem in a plane domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with smooth boundary <inline-formula content-type="math/mathml"> <mml:math … Consider Dirichlet’s problem in a plane domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with smooth boundary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential normal upper Omega"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial \Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For the purpose of its approximate solution, an approximating domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h Baseline comma 0 greater-than h less-than-over-equals 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>h</mml:mi> <mml:mo>≦</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h},0 > h \leqq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with polygonal boundary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential normal upper Omega Subscript h"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial {\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is introduced where the segments of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential normal upper Omega Subscript h"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial {\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have length at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A projection method introduced by Nitsche [6] is then applied on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to give an approximate solution in a finite-dimensional subspace of functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{S_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for instance a space of splines defined on a triangulation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The boundary terms in the bilinear form associated with Nitsche’s method are modified to correct for the perturbation of the boundary.

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Error estimates for a Galerkin method for a class of model equations for long waves

1974-08-01

Lars B. Wahlbin

Galerkin methods applied to the Benjamin-Bona-Mahony equation

1975-03-01

Marco Antonio Raupp

A priori estimates for the solutions of difference approximations to parabolic partial differential equations

1960-09-01

Milton Lees

$L_\infty $ Estimates of Optimal Orders for Galerkin Methods for One-Dimensional Second Order Parabolic and Hyperbolic Equations

1973-10-01

Mary F. Wheeler

A priori $L_\infty $ error estimates are derived for continuous piecewise polynomial Galerkin approximations to the solutions of one-dimensional second order parabolic and hyperbolic equations. Optimal rates of convergence are … A priori $L_\infty $ error estimates are derived for continuous piecewise polynomial Galerkin approximations to the solutions of one-dimensional second order parabolic and hyperbolic equations. Optimal rates of convergence are established for both continuous and discrete time Galerkin procedures.