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A Multilevel Method for Discontinuous Galerkin Approximation of Threedimensional Elliptic Problems
"... Summary. We construct optimal order multilevel preconditioners for interiorpenalty discontinuous Galerkin (DG) finite element discretizations of 3D elliptic boundaryvalue problems. A specific assembling process is proposed which allows us to characterize the hierarchical splitting locally. This is ..."
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Summary. We construct optimal order multilevel preconditioners for interiorpenalty discontinuous Galerkin (DG) finite element discretizations of 3D elliptic boundaryvalue 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 twolevel 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. 1
MULTILEVEL PRECONDITIONING OF ELLIPTIC PROBLEMS DISCRETIZED BY A CLASS OF DISCONTINUOUS GALERKIN METHODS
"... Abstract. We present optimal order preconditioners for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems. A specific assembling process is proposed which allows us to use the hierarchy of geometrically nested meshes. We consider two variants of hi ..."
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Abstract. We present optimal order preconditioners for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems. A specific assembling process is proposed which allows us to use the hierarchy of geometrically nested meshes. We consider two variants of hierarchical splittings and study the angle between the resulting subspaces. Applying the corresponding twolevel basis transformation recursively a sequence of algebraic problems is generated that can be associated with a hierarchy of coarse versions of DG approximations of the original problem. New bounds for the constant γ in the strengthened CauchyBunyakowskiSchwarz inequality are derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Key words. Discontinuous Galerkin FEM, multilevel preconditioning, hierarchical basis, CBS constant AMS subject classifications. 65N30, 65N22, 65N55