The gradient discretisation method (GDM) – a generic framework encompassing many numerical methods – is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical …
The gradient discretisation method (GDM) – a generic framework encompassing many numerical methods – is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorokhod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework.
The gradient discretisation method (GDM) -- a generic framework encompassing many numerical methods -- is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical …
The gradient discretisation method (GDM) -- a generic framework encompassing many numerical methods -- is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorohod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework.
The gradient discretisation method (GDM) – a generic framework encompassing many numerical methods – is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical …
The gradient discretisation method (GDM) – a generic framework encompassing many numerical methods – is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorokhod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework.
The gradient discretisation method (GDM) -- a generic framework encompassing many numerical methods -- is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical …
The gradient discretisation method (GDM) -- a generic framework encompassing many numerical methods -- is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorohod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework.
It is shown that in a large class of topological spaces every uniformly tight sequence of random elements contains a subsequence which admits the usual almost sure (a.s.) Skorokhod representation …
It is shown that in a large class of topological spaces every uniformly tight sequence of random elements contains a subsequence which admits the usual almost sure (a.s.) Skorokhod representation on the Lebesgue interval.
The paper deals with stochastic partial differential equations driven by Poisson random measures of jump type and their numerical approximation. We investigate the accuracy of space and time approximation. As …
The paper deals with stochastic partial differential equations driven by Poisson random measures of jump type and their numerical approximation. We investigate the accuracy of space and time approximation. As space approximation we consider finite elements and as time approximation the implicit Euler scheme. A corrected version of this paper has been appended to the originally posted pdf.
We show in this paper that the gradient schemes (which encompass a large family of discrete schemes) may be used for the approximation of the Stefan problem $\partial_t \bar u …
We show in this paper that the gradient schemes (which encompass a large family of discrete schemes) may be used for the approximation of the Stefan problem $\partial_t \bar u - \Delta \zeta (\bar u) = f$. The convergence of the gradient schemes to the continuous solution of the problem is proved thanks to the following steps. First, estimates show (up to a subsequence) the weak convergence to some function $u$ of the discrete function approximating $\bar u$. Then Alt-Luckhaus' method, relying on the study of the translations with respect to time of the discrete solutions, is used to prove that the discrete function approximating $\zeta(\bar u)$ is strongly convergent (up to a subsequence) to some continuous function $\chi$. Thanks to Minty's trick, we show that $\chi = \zeta(u)$. A convergence study then shows that $u$ is then a weak solution of the problem, and a uniqueness result, given here for fitting with the precise hypothesis on the geometric domain, enables to conclude that $u = \bar u$. This convergence result is illustrated by some numerical examples using the Vertex Approximate Gradient scheme.
Moving boundary problems allow to model systems with phase transition at an inner boundary. Motivated by problems in economics and finance, we set up a price-time continuous model for the …
Moving boundary problems allow to model systems with phase transition at an inner boundary. Motivated by problems in economics and finance, we set up a price-time continuous model for the limit order book and consider a stochastic and nonlinear extension of the classical Stefan-problem in one space dimension. Here, the paths of the moving interface might have unbounded variation, which introduces additional challenges in the analysis. Working on the distribution space, the Itô–Wentzell formula for SPDEs allows to transform these moving boundary problems into partial differential equations on fixed domains. Rewriting the equations into the framework of stochastic evolution equations and stochastic maximal $L^{p}$-regularity, we get existence, uniqueness and regularity of local solutions. Moreover, we observe that explosion might take place due to the boundary interaction even when the coefficients of the original problem have linear growths.
Abstract The gradient discretization method (GDM) is a generic framework, covering many classical methods (finite elements, finite volumes, discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. …
Abstract The gradient discretization method (GDM) is a generic framework, covering many classical methods (finite elements, finite volumes, discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. In this paper we study the GDM for a general stochastic evolution problem based on a Leray–Lions type operator. The problem contains the stochastic $p$-Laplace equation as a particular case. The convergence of the gradient scheme (GS) solutions is proved by using discrete functional analysis techniques, Skorohod theorem and the Kolmogorov test. In particular, we provide an independent proof of the existence of weak martingale solutions for the problem. In this way we lay foundations and provide techniques for proving convergence of the GS approximating stochastic partial differential equations.
Abstract We study the stochastic p -Laplace system in a bounded domain. We propose two new space–time discretizations based on the approximation of time-averaged values. We establish linear convergence in …
Abstract We study the stochastic p -Laplace system in a bounded domain. We propose two new space–time discretizations based on the approximation of time-averaged values. We establish linear convergence in space and 1/2 convergence in time. Additionally, we provide a sampling algorithm to construct the necessary random input in an efficient way. The theoretical error analysis is complemented by numerical experiments.