A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation. (20th May 2021)

• Record Type: Journal Article
• Authors: Benner, Peter; Stillfjord, Tony; Trautwein, Christoph
• Is Part Of: IMA journal of numerical analysis. Volume 42:Number 3(2022)
• Page Start: 2118
• Page End: 2150
• Publication Date: 2021-05-20
• Subjects: optimal control -- stochastic partial differential equation -- full discretization -- convergence analysis -- heat conduction; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drab033
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous 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.