Domain decomposition strategies for the stochastic heat equation

Abstract

We consider the numerical approximation of mild solutions of the stochastic, Hilbert space-valued heat equation

with a uniformly elliptic operator A:D(A)→L ₂(𝒟) and a symmetric operator Q:L ₂(𝒟)→L ₂(𝒟) with finite trace. We apply different domain decomposition algorithms based on explicit and implicit time-stepping, together with a finite element and backward Euler discretization to solve the problem, and derive optimal strong and weak rates of convergence. For this purpose, and due to the interplay of limited regularity in time of the driving noise and the splitting character of the scheme, a well-known deterministic domain decomposition algorithm requires modifications to prove an optimal weak rate of convergence.

Keywords:

• weak order
• strong order
• stochastic heat equation
• finite element
• Euler scheme
• domain decomposition

2010 AMS Subject Classifications:

• 60H15
• 65M55
• 65M60
• 65M15