Analysis Of A Domain Decomposition-Based Cell-Centered Method for Heterogeneous Anisotropic Diffusion Problems

TL;DR

This paper develops and analyzes a domain decomposition method for a cell-centered finite element scheme tailored for heterogeneous, anisotropic diffusion problems, showing convergence independent of mesh size and coefficient jumps.

Contribution

It introduces a domain decomposition approach for the FECC scheme with a tailored Neumann-Neumann preconditioner and proves convergence of the associated iterative algorithm.

Findings

Preconditioned iterative algorithm converges independently of mesh size.

Method effectively handles coefficient jumps in heterogeneous media.

Numerical results confirm theoretical convergence properties.

Abstract

The paper is concerned with the derivation and analysis of nonoverlapping domain decomposition for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. However, the substructuring procedure as well as the Neumann-Neumann type preconditioner can be adapted to the domain decomposition-based FECC method naturally. Convergence analysis of a preconditioned iterative algorithm, namely the Dirichlet-Neumann to Neumann-Neumann algorithm, associated with the discrete FECC interface problem is the main focus of this work. Two dimensional numerical results for two subdomains with conforming meshes demonstrate that the preconditioned iterative algorithm converges independently of the mesh size and the coefficient jump.