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A multilevel method for discontinuous Galerkin approximation of three‐dimensional anisotropic elliptic problems
Author(s) -
Kraus J. K.,
Tomar S. K.
Publication year - 2008
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.544
Subject(s) - mathematics , discontinuous galerkin method , linear subspace , finite element method , polygon mesh , multigrid method , galerkin method , mathematical analysis , geometry , partial differential equation , physics , thermodynamics
We construct optimal order multilevel preconditioners for interior‐penalty discontinuous Galerkin (DG) finite element discretizations of three‐dimensional (3D) anisotropic elliptic boundary‐value problems. In this paper, we extend the analysis of our approach, introduced earlier for 2D problems ( SIAM J. Sci. Comput. , accepted), to cover 3D 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. A new bound for the constant γ in the strengthened Cauchy–Bunyakowski–Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Copyright © 2007 John Wiley & Sons, Ltd.