Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jump coefficients

Abstract

SUMMARY 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 rates of the (multiplicative) two-grid and multigrid V-cycle algorithms will deteriorate rapidly because of large jumps in coefficient. However, the preconditioned systems have only a fixed number of small eigenvalues depending on the large jump in coefficient, and the effective condition numbers are independent of the coefficient and bounded logarithmically with respect to the mesh size. As a result, the two-grid or multigrid preconditioned conjugate gradient algorithm converges nearly uniformly. We also comment on some major differences of the convergence theory between the nonconforming case and the standard conforming case. Numerical experiments support the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.