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A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients. (English) Zbl 1306.65178

Summary: In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart (C-R) finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted \(L ^{2}\) norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.

MSC:

65F08 Preconditioners for iterative methods
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35R05 PDEs with low regular coefficients and/or low regular data
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
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