Additive average Schwarz method for a Crouzeix-Raviart Finite Volume Element Discretization of Elliptic Problems with Heterogeneous Coefficients

TL;DR

This paper develops an additive Schwarz method tailored for a Crouzeix-Raviart Finite Volume Element discretization of elliptic problems with heterogeneous coefficients, analyzing its convergence behavior relative to mesh and coefficient distribution.

Contribution

It introduces a new additive Schwarz preconditioner for CRFVE discretizations addressing discontinuous coefficients and analyzes its convergence properties.

Findings

Convergence rate depends linearly or quadratically on mesh parameters H/h.

Under certain conditions, convergence is independent of coefficient jumps.

The method improves solver efficiency for problems with complex coefficient distributions.

Abstract

In this paper we introduce an additive Schwarz method for a Crouzeix-Raviart Finite Volume Element (CRFVE) discretization of a second order elliptic problem with discontinuous coefficients, where the discontinuities are both inside the subdomains and across and along the subdomain boundaries. We show that, depending on the distribution of the coefficient in the model problem, the parameters describing the GMRES convergence rate of the proposed method depend linearly or quadratically on the mesh parameters $H/h$. Also, under certain restrictions on the distribution of the coefficient, the convergence of the GMRES method is independent of jumps in the coefficient.