Multigrid Preconditioner for Nonconforming Discretization of Elliptic Problems with Jump Coefficients
In this paper, we present a multigrid preconditioner for solving the linear system arising from the piecewise linear nonconforming Crouzeix-Raviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses the standard conforming subspaces as coarse spaces. Numerical tests show both robustness with respect to the jump in the coefficient and near-optimality with respect to the number of degrees of freedom.
Unable to display preview. Download preview PDF.
- .B. Ayuso de Dios, M. Holst, Y. Zhu, and L. Zikatanov. Multilevel Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems with Jump Coefficients. Arxiv preprint arXiv:1012.1287, to appear in Mathematics of Computation, 2010.Google Scholar
- .J. H. Bramble. Multigrid Methods, volume 294 of Pitman Research Notes in Mathematical Sciences. Longman Scientific & Technical, Essex, England, 1993.Google Scholar
- .W. L. Briggs, V. E. Henson, and S. F. McCormick. A multigrid tutorial. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2000.Google Scholar
- .M. Dryja, M. V. Sarkis, and Olof B. Widlund. Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numerische Mathematik, 72(3):313–348, 1996.Google Scholar
- .M. Sarkis. Multilevel methods for P 1 nonconforming finite elements and discontinuous coefficients in three dimensions. In Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993), volume 180 of Contemp. Math., pages 119–124. Amer. Math. Soc., Providence, RI, 1994.Google Scholar
- .M. V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute of Mathematical Science of New York University, 1994.Google Scholar
- .J. Xu. Theory of Multilevel Methods. PhD thesis, Cornell University, 1989.Google Scholar
- .Y. Zhu. Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic PDE with jump coefficient. Arxiv preprint arXiv:1110.5159, Numerical Linear Algebra with Applications, DOI: 10.1002/nla.1856, 2012.Google Scholar