A Domain Decomposition Method for Stochastic Evolution Equations

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2611-2639, December 2024.
Abstract. In recent years, stochastic partial differential equations (SPDEs) have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes provide powerful, efficient, and flexible numerical methods for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where one splits the domain into subdomains and constructs the numerical approximation in a divide-and-conquer strategy. Instead of solving one expensive problem on the entire domain, one then deals with cheaper problems on the subdomains. This is particularly useful in modern computer architectures, as the subproblems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. This implies that the existing convergence analysis is not directly applicable, even though the building blocks of the operator splitting domain decomposition method are standard. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments.