On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs

TL;DR

This paper analyzes the convergence of an inexact implicit Euler scheme for SPDEs, focusing on how adaptive spatial discretization errors affect overall convergence and how to choose tolerances for optimal accuracy.

Contribution

It provides a theoretical framework for understanding error propagation in adaptive spatial discretizations within the implicit Euler scheme for SPDEs.

Findings

Error propagation in adaptive spatial discretization is characterized.

Guidelines for choosing tolerances to maintain convergence order.

The scheme achieves optimal convergence with properly controlled inexact solves.

Abstract

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.