An optimal order multilevel preconditioner with respect to problem and discretization parameters

  • Authors:

  • Owe Axelsson;Svetozar Margenov

  • Affiliations:

  • Subfaculty of Mathematics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands;Central Laboratory of Parallel Processing, Bulgarian Academy of Sciences , Acad. G. Bontchev str., Bl. 25A, 1113 Sofia, Bulgaria

  • Venue:
  • Scientific computing and applications
  • Year:
  • 2001

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Abstract

Preconditioners based on various multilevel extensions of two-level finite element methods lead to iterative methods which have an optimal order computational complexity with respect to the size (or discretization parameter) of the system. The methods can be on block matrix factorized form, recursively extended via certain matrix polynomial approximations of the arising Schur complement matrices or on additive, i.e. block diagonal form using stabilizations of the condition number at certain levels. The resulting spectral equivalence holds uniformly with respect to jumps in the coefficients of the differential operator and for arbitrary triangulations. Such methods were first presented by Axelsson and Vassilevski in the late 80s.An important part of the algorithm is the treatment of the systems with the diagonal block matrix, which arise on each finer level and corresponds to the added degrees of freedom on that level. This block is well-conditioned for model type problems but becomes increasingly ill-conditioned when the coefficient matrix becomes more anisotropic or, equivalently, when the mesh aspect ratio increases.In the paper two methods are presented to approximate this matrix also leading to a preconditioner with spectral equivalence bounds which hold uniformly with respect to both the problem and discretization parameters. The same holds therefore also for the preconditioner to the global matrix.