Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem

Abstract

The purpose of this paper is to establish a new numerical approach to solve, in two dimensions, a semilinear reaction diffusion equation combining finite volume method and Schur complement method. We applied our method for q = 2 non-overlapping subdomains and then we generalized in the case of several subdomains (q ≥ 2). A large number of numerical test cases shows the efficiency and the good accuracy of the proposed approach in terms of the CPU time and the order of the error, when increasing the number of subdomains, without using the parallel computing. After several variations of the number of subdomains and the mesh grid, we remark two significant results. On the one hand, the increase related to the number of subdomains does not affect the order of the error, on the other hand, for each mesh grid when we augment the number of subdomains, the CPU time reaches the minimum for a specific number of subdomains. In order to have the minimum CPU time, we resorted to a statistical study between the optimal number of subdomains and the mesh grid.

Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique

“…This DDM allows to compute solutions using interfacial system known as the Schur complement system. It was studied in several works, for example in [4] and [15], it has been studied for second order elliptic problems using conforming finite elements, in [6] for elliptic boundary value problems of 2m order for conforming and nonconforming finite elements, in [7] for the biharmonic equation on a rectangle for finite difference scheme, and in [3] for a semilinear reaction diffusion problem using FV. This version of Schur complement method is more expensive when the number of unknowns on the interface increase.…”

“…we add a new unknowns on the interface, and we reformulate the global system into a block system, see e.g. [3].…”