Consistency and Stability of a Milstein-Galerkin Finite Element Scheme for Semilinear SPDE

TL;DR

This paper introduces an abstract error analysis framework for semilinear SPDEs and proves strong convergence of a Milstein-Galerkin finite element scheme, including error estimates and extensions to truncated noise.

Contribution

It generalizes the concept of bistability to the semigroup framework and establishes strong convergence and error bounds for the Milstein-Galerkin scheme.

Findings

Proved strong convergence of the Milstein-Galerkin scheme.

Derived two-sided error estimates for spatio-temporal discretization.

Extended results to schemes with truncated noise.

Abstract

We present an abstract concept for the error analysis of numerical schemes for semilinear stochastic partial differential equations (SPDEs) and demonstrate its usefulness by proving the strong convergence of a Milstein-Galerkin finite element scheme. By a suitable generalization of the notion of bistability from Beyn & Kruse (DCDS B, 2010) to the semigroup framework in Hilbert spaces, our main result includes a two-sided error estimate of the spatio-temporal discretization. In an additional section we derive an analogous result for a Milstein-Galerkin finite element scheme with truncated noise.