A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation

TL;DR

This paper develops and analyzes a finite element and implicit Euler method for approximating solutions to a controlled stochastic heat equation with boundary noise, achieving optimal convergence under certain regularity conditions.

Contribution

It introduces a combined finite element and implicit Euler scheme for a stochastic heat control problem with boundary noise, proving optimal convergence orders under smoothness assumptions.

Findings

Optimal convergence orders are achieved for smooth data.

Convergence order decreases with less smooth data.

Numerical experiments confirm theoretical results.

Abstract

We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with non-homogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above 1/2 in space and 1/4 in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions, the convergence order is further decreased. Our results only assume that the related (deterministic) differential Riccati equation can be approximated with a certain convergence order, which is easy to achieve in practice. We confirm these theoretical results through a numerical experiment in a two dimensional domain.