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Additive Schwarz Preconditioners for Degenerate Problems with Isotropic Coefficients

In: Numerical Mathematics and Advanced Applications

Author

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  • S. Beuchler

    (JKU Linz, Institute of Computational Mathematics)

Abstract

This paper deals with the numerical solution of the degenerate elliptic boundary value problem $$-\nabla \cdot x^{\alpha}\nabla u(x,y)=f(x,y), \alpha \geq 0$$ in Ω = (0, 1)2. This boundary value problem is discretized by piecewise linear finite elements on regular Cartesian grids. The corresponding linear system of algebraic equations is solved by a preconditioned conjugate gradient method with the Bramble-Pasciak-Xu preconditioner. A uniform bound of the condition number of the preconditioned system matrix is proved for α ≠ 1. The proof makes use of another additive Schwarz splitting arising from an overlapping domain decomposition of the computational domain Ω. Some numerical experiments show the efficiency of the proposed algorithm also for general polygonal domains.

Suggested Citation

  • S. Beuchler, 2008. "Additive Schwarz Preconditioners for Degenerate Problems with Isotropic Coefficients," Springer Books, in: Karl Kunisch & Günther Of & Olaf Steinbach (ed.), Numerical Mathematics and Advanced Applications, pages 143-150, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-69777-0_16
    DOI: 10.1007/978-3-540-69777-0_16
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