Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic PDE with jump coefficient
This work addresses the challenge of solving linear systems from nonconforming finite element methods for elliptic problems with highly varying coefficients, providing a robust preconditioner for practitioners.
The paper analyzes a multigrid V-cycle preconditioner for Crouzeix-Raviart discretization of elliptic PDEs with jump coefficients, showing that while the convergence rate of the V-cycle alone deteriorates with large jumps, the preconditioned conjugate gradient method converges nearly uniformly with an effective condition number bounded logarithmically in mesh size.
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.