Subdomain solution decomposition method for nonstationary problems

TL;DR

This paper introduces a novel domain decomposition method for nonstationary problems that decomposes the solution itself rather than the operators, aiming to reduce computational costs in solving parabolic equations.

Contribution

The paper develops a new subdomain solution decomposition approach for nonstationary problems, extending previous operator-based methods to improve computational efficiency.

Findings

Effective domain decomposition schemes for parabolic equations

Comparison of overlapping and non-overlapping subdomain methods

Numerical results validate theoretical convergence and efficiency

Abstract

The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in time. The traditional construction approach of splitting schemes is based on an additive representation of the problem operator(s) and uses explicit-implicit approximations for individual terms. Recently (Y. Efendiev, P.N. Vabishchevich. Splitting methods for solution decomposition in nonstationary problems. \textit{Applied Mathematics and Computation}. \textbf{397}, 125785, 2021), a new class of methods of approximate solution of nonstationary problems has been introduced based on decomposition not of operators but of the solution itself. This new approach with subdomain solution selection is used in this paper to construct domain decomposition schemes. The boundary value problem for a second-order parabolic equation in a rectangle with a difference approximation in space is typical. Two and three-level schemes for decomposition of the domain with and without overlapping subdomains are investigated. Our numerical experiments complement the theoretical results.