Explicit Extension Operators on Hierarchical Grids

Extension operators extend functions deened on the boundary of a domain into its interior. This paper presents explicit extension operators by means of multilevel decompositions on hierarchical grids. It is shown that the norm-preserving property of these operators holds for the 2D as well for the 3D case with constants independent on dis-cretization and domain size. These constants can be further improved by an additional iteration scheme applied to the extension operator. Some implementation of these techniques is presented for a domain decomposition preconditioner and numerical experiments are given.

[1]  Stability of finite-difference schemes for elliptic equations with respect to Dirichlet boundary conditions , 1972 .

[2]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .

[3]  Harry Yserentant,et al.  On the multi-level splitting of finite element spaces , 1986 .

[4]  A. M. MATSOKIN,et al.  Norms in the space of traces of mesh functions , 1988 .

[5]  J. Pasciak,et al.  Parallel multilevel preconditioners , 1990 .

[6]  Harry Yserentant,et al.  Two preconditioners based on the multi-level splitting of finite element spaces , 1990 .

[7]  S. V. NEPOMNYASCHIKH.  Method of splitting into subspaces for solving elliptic boundary value problems in complex-form domains , 1991 .

[8]  S. Nepomnyaschikh.  Mesh theorems on traces, normalizations of function traces and their inversion , 1991 .

[9]  The non-overlapping domain decomposition multiplicative schwarz method , 1992 .

[10]  W. Dahmen,et al.  Multilevel preconditioning , 1992 .

[11]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[12]  Hsuanjen Cheng.  Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of , 1993 .

[13]  Harry Yserentant,et al.  A basic norm equivalence for the theory of multilevel methods , 1993 .

[14]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .

[15]  Arnd Meyer,et al.  Hierarchical Extension Operators and Local Multigrid Methods in Domain Decomposition Preconditioners , 1994 .

[16]  Sergey V. Nepomnyaschikh.  Optimal Multilevel Extension Operators , 1995 .

[17]  Multilevel Extension Techniques in Domain Decomposition Preconditioners , 1996 .

[18]  Michael Jung,et al.  Parallel Solvers for Nonlinear Elliptic Problems Based on Domain Decomposition Ideas , 1997, Parallel Comput..

[19]  Gundolf Haase.  Hierarchical extension operators plus smoothing in domain decomposition preconditioners , 1997 .

[20]  Panayot S. Vassilevski.  On Some Applications of the ...-Stable Wavelet-Like Hierarchical Finite Element Space Decompositions , 1998 .