Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

Maria Lymbery; Svetozar Margenov

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 357-369
  • ISSN: 2391-5455

Abstract

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While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.

How to cite

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Maria Lymbery, and Svetozar Margenov. "Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems." Open Mathematics 10.1 (2012): 357-369. <http://eudml.org/doc/269572>.

@article{MariaLymbery2012,
abstract = {While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.},
author = {Maria Lymbery, Svetozar Margenov},
journal = {Open Mathematics},
keywords = {Biquadratic finite elements; Multilevel methods; Robust preconditioning; biquadratic finite elements; multilevel methods; robust preconditioning; numerical examples; inequality; semi-coarsening mesh refinement; algebraic multi level iteration; Cauchy-Bunyakovsky-Schwarz; algorithm},
language = {eng},
number = {1},
pages = {357-369},
title = {Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems},
url = {http://eudml.org/doc/269572},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Maria Lymbery
AU - Svetozar Margenov
TI - Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 357
EP - 369
AB - While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.
LA - eng
KW - Biquadratic finite elements; Multilevel methods; Robust preconditioning; biquadratic finite elements; multilevel methods; robust preconditioning; numerical examples; inequality; semi-coarsening mesh refinement; algebraic multi level iteration; Cauchy-Bunyakovsky-Schwarz; algorithm
UR - http://eudml.org/doc/269572
ER -

References

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