Robust AMLI methods for parabolic Crouzeix-Raviart FEM systems

  • Authors:

  • Petia Boyanova;Svetozar Margenov;Maya Neytcheva

  • Affiliations:

  • Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 25A, 1113 Sofia, Bulgaria and Division of Scientific Computing, Department of Information Technology, U ...;Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 25A, 1113 Sofia, Bulgaria;Division of Scientific Computing, Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2010

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Abstract

For the iterative solution of linear systems of equations arising from finite element discretization of elliptic problems there exist well-established techniques to construct numerically efficient and computationally optimal preconditioners. Among those, most often preferred choices are Multigrid methods (geometric or algebraic), Algebraic MultiLevel Iteration (AMLI) methods, Domain Decomposition techniques. In this work, the method in focus is AMLI. We extend its construction and the underlying theory over to systems arising from discretizations of parabolic problems, using non-conforming finite element methods (FEM). The AMLI method is based on an approximated block two-by-two factorization of the original system matrix. A key ingredient for the efficiency of the AMLI preconditioners is the quality of the utilized block two-by-two splitting, quantified by the so-called Cauchy-Bunyakowski-Schwarz (CBS) constant, which measures the abstract angle between the two subspaces, associated with the two-by-two block splitting of the matrix. The particular choice of space discretization for the parabolic equations, used in this paper, is Crouzeix-Raviart non-conforming elements on triangular meshes. We describe a suitable splitting of the so-arising matrices and derive estimates for the associated CBS constant. The estimates are uniform with respect to discretization parameters in space and time as well as with respect to coefficient and mesh anisotropy, thus providing robustness of the method.