Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems

TL;DR

This paper introduces symmetric and nonsymmetric additive Schwarz preconditioners for finite volume element discretizations of symmetric elliptic problems, demonstrating their robustness and weak dependence on mesh parameters and coefficient jumps.

Contribution

It proposes new additive Schwarz preconditioners tailored for finite volume element discretizations, with proven convergence properties independent of coefficient jumps.

Findings

Convergence depends polylogarithmically on mesh parameters.

Preconditioners are robust against large coefficient jumps.

Weak dependence on mesh parameters enhances efficiency.

Abstract

A symmetric and a nonsymmetric variant of the additive Schwarz preconditioner are proposed for the solution of a nonsymmetric system of algebraic equations arising from a general finite volume element discretization of symmetric elliptic problems with large jumps in the entries of the coefficient matrices across subdomains. It is shown in the analysis, that the convergence of the preconditioned GMRES iteration with the proposed preconditioners, depends polylogarithmically on the mesh parameters, in other words, the convergence is only weakly dependent on the mesh parameters, and it is robust with respect to the jumps in the coefficients.