We construct optimal order multilevel preconditioners for interiorpenalty discontinuous Galerkin (DG) finite element discretizations of 3D elliptic boundary-value problems. A specific assembling process is proposed which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. The presented numerical results demonstrate the potential of this approach.
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References
D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779, 2001/02.
O. Axelsson. Iterative Solution Methods. Cambridge University Press, Cambridge, 1994.
O. Axelsson and P.S. Vassilevski. Algebraic multilevel preconditioning methods. I. Numer. Math., 56(2–3):157–177, 1989.
O. Axelsson and P.S. Vassilevski. Algebraic multilevel preconditioning methods. II. SIAM J. Numer. Anal., 27(6):1569–1590, 1990.
O. Axelsson and P.S. Vassilevski. Variable-step multilevel preconditioning methods. I. Selfadjoint and positive definite elliptic problems. Numer. Linear Algebra Appl., 1(1):75–101, 1994.
S.C. Brenner and J. Zhao. Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math., 2(1):3–18, 2005.
V.A. Dobrev, R.D. Lazarov, P.S. Vassilevski, and L.T. Zikatanov. Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations. Numer. Linear Algebra Appl., 13(9):753–770, 2006.
V. Eijkhout and P.S. Vassilevski. The role of the strengthened Cauchy-BuniakowskiÄ-Schwarz inequality in multilevel methods. SIAM Rev., 33(3):405–419, 1991.
J. Gopalakrishnan and G. Kanschat. A multilevel discontinuous Galerkin method. Numer. Math., 95(3):527–550, 2003.
J.K. Kraus. An algebraic preconditioning method for M-matrices: linear versus non-linear multilevel iteration. Numer. Linear Algebra Appl., 9(8):599–618, 2002.
J.K. Kraus and S.K. Tomar. A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems. Numer. Linear Algebra Appl., in press.
J.K. Kraus and S.K. Tomar. Multilevel preconditioning of two-dimensional elliptic problems discretized by a class of discontinuous Galerkin methods. SIAM J. Sci. Comput., to appear.
L. Lazarov and S. Margenov. CBS constants for graph-Laplacians and application to multilevel methods for discontinuous Galerkin systems. J. Complexity, doi: 10.1016/j.jco.2006.10.003, 2006.
Y. Saad. Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, second edition, 2003.
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Kraus, J.K., Tomar, S.K. (2008). A Multilevel Method for Discontinuous Galerkin Approximation of Three-dimensional Elliptic Problems. In: Langer, U., Discacciati, M., Keyes, D.E., Widlund, O.B., Zulehner, W. (eds) Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75199-1_14
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