Optimal preconditioning for Raviart-Thomas mixed formulation of second-order elliptic problems

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    Abstract

    We evaluate two preconditioning strategies for the indefinite linear system obtained from Raviart-Thomas mixed finite element formulation of a second-order elliptic problem with variable diffusion coefficients. It is known that the underlying saddle-point problem is well-posed in two function spaces, H(div) × L2and L2× H1, leading to the possibility of two distinct types of preconditioner. For homogeneous Dirichlet boundary conditions, the discrete problems are identical. This motivates our use of Raviart-Thomas approximation in both frameworks, yielding a nonconforming method in the second case. The focus is on linear algebra; we establish the optimality of two parameter-free block-diagonal preconditioners using basic properties of the finite element matrices. Uniform eigenvalue bounds are established and the impact of the PDE coefficients is explored in numerical experiments. A practical scheme is discussed, the key building block for which is a fast solver for a scalar diffusion operator based on algebraic multigrid. Trials of preconditioned MINRES illustrate that both preconditioning schemes are optimal with respect to the discretization parameter and robust with respect to the PDE coefficients.
    Original languageEnglish
    Pages (from-to)718-738
    Number of pages21
    JournalSIAM Journal on Matrix Analysis and Applications
    Volume25
    Issue number3
    DOIs
    Publication statusPublished - 2003

    Keywords

    • MINRES
    • Mixed finite elements
    • Preconditioning
    • Raviart-Thomas
    • Saddle-point problems
    • Second-order elliptic problems
    • Variable coefficients

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