Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic PDE with jump coefficient

TL;DR

This paper develops a multigrid V-cycle preconditioner for nonconforming Crouzeix-Raviart discretizations of elliptic PDEs with jump coefficients, demonstrating nearly uniform convergence and robustness through theoretical analysis and numerical tests.

Contribution

It introduces a multigrid preconditioner that remains effective despite large coefficient jumps, with proven bounded condition number and near-uniform convergence.

Findings

Convergence rate deteriorates with large coefficient jumps.

Preconditioned system has a fixed number of small eigenvalues.

Numerical tests confirm robustness against coefficient jumps and mesh size.

Abstract

In this paper, we present a multigrid $V$-cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix-Raviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rate of the multigrid $V$-cycle algorithm will deteriorate rapidly due to large jumps in coefficient. However, the preconditioned system has only a fixed number of small eigenvalues, which are deteriorated due to the large jump in coefficient, and the effective condition number is bounded logarithmically with respect to the mesh size. As a result, the multigrid $V$-cycle preconditioned conjugate gradient algorithm converges nearly uniformly. Numerical tests show both robustness with respect to jumps in the coefficient and the mesh size.