Edge based Schwarz methods for the Crouzeix-Raviart finite volume element discretization of elliptic problems

TL;DR

This paper introduces two variants of the Additive Schwarz Method tailored for Crouzeix-Raviart finite volume discretizations of elliptic problems with discontinuous coefficients, achieving near-optimal convergence rates.

Contribution

The paper develops and analyzes two new Schwarz preconditioners for CRFVE discretizations, demonstrating their near-optimal performance with residual error estimates depending polylogarithmically on mesh parameters.

Findings

Preconditioners are almost optimal for the discretization.

Residual error estimates depend polylogarithmically on mesh parameters.

Both symmetric and nonsymmetric variants are effective.

Abstract

In this paper, we present two variants of the Additive Schwarz Method for a Crouzeix-Raviart finite volume element (CRFVE) discretization of second order elliptic problems with discontinuous coefficients where the discontinuities are only across subdomain boundaries. One preconditioner is symmetric while the other is nonsymmetric. The proposed methods are almost optimal, in the sense that the residual error estimates for the GMRES iteration in the both cases depend only polylogarithmically on the mesh parameters.