Quadrature-Galerkin approximations to solutions of elliptic differential equations

Abstract

In practice the Galerkin method for solving elliptic partial differential equations yields equations involving certain integrals which cannot be evaluated analytically. Instead these integrals are approximated numerically and the resulting equations are solved to give “quadrature-Galerkin approximations” to the solution of the differential equation. Using a technique of J. Nitsche, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a priori error bounds are obtained for the difference between the solution of the differential equation and a class of quadrature-Galerkin approximations.

Locations

• Proceedings of the American Mathematical Society - View - PDF