Stabilizing the Hierarchical Basis by Approximate Wavelets, I: Theory

This paper proposes a stabilization of the classical hierarchical basis (HB) method by modifying the HB functions using some computationally feasible approximate L2-projections onto finite element spaces of relatively coarse levels. The corresponding multilevel additive and multiplicative algorithms give spectrally equivalent preconditioners, and one action of such a preconditioner is of optimal order computationally. The results are regularity-free for the continuous problem (second order elliptic) and can be applied to problems with rough coefficients and local refinement. © 1997 by John Wiley & Sons, Ltd.

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

[2]  P. Oswald,et al.  Remarks on the Abstract Theory of Additive and Multiplicative Schwarz Algorithms , 1995 .

[3]  S. Jaffard.  Wavelet methods for fast resolution of elliptic problems , 1992 .

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

[5]  J. Pasciak,et al.  Convergence estimates for product iterative methods with applications to domain decomposition , 1991 .

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

[7]  R. Bank,et al.  The hierarchical basis multigrid method , 1988 .

[8]  H. Yserentant.  Old and new convergence proofs for multigrid methods , 1993, Acta Numerica.

[9]  P. Vassilevski.  Hybrid V-cycle algebraic multilevel preconditioners , 1992 .

[10]  Panayot S. Vassilevski,et al.  An application of the abstract multilevel theory to nonconforming finite element methods , 1995 .

[11]  Rob Stevenson,et al.  A robust hierarchical basis preconditioner on general meshes , 1997 .

[12]  Michael Griebel,et al.  Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems , 1995, Adv. Comput. Math..

[13]  D. Braess.  The contraction number of a multigrid method for solving the Poisson equation , 1981 .

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

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

[16]  Michael Griebel,et al.  Multilevel Algorithms Considered as Iterative Methods on Semidefinite Systems , 1994, SIAM J. Sci. Comput..

[17]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[18]  Panayot S. Vassilevski,et al.  On Two Ways of Stabilizing the Hierarchical Basis Multilevel Methods , 1997, SIAM Rev..

[19]  Joseph E. Pasciak,et al.  Uniform convergence estimates for multigrid V-cycle algorithms with less than full elliptic regularity , 1992 .

[20]  O. Axelsson,et al.  Algebraic multilevel preconditioning methods, II , 1990 .

[21]  I. Gustafsson,et al.  Preconditioning and two-level multigrid methods of arbitrary degree of approximation , 1983 .