Domain Decomposition and Iterative Refinement Methods for Mixed Finite Element Discretisations of Elliptic Problems (Classic Reprint)

Excerpt from Domain Decomposition and Iterative Refinement Methods for Mixed Finite Element Discretisations of Elliptic Problems

In this Chapter, we present the relevent background about saddle point formula tions of elliptic problems, and mixed finite element methods using the raviart-thomas spaces. We then discuss an extension theorem for the raviart-thomas finite element spaces, which we use later in establishing bounds for the rate of convergence of the domain decomposition methods. A section containing some background on iterative methods, is also included.

In Chapter 2, we discuss algorithms involving subdomains with overlap, such as the classical Schwarz alternating method, cf. Lions and the additive Schwarz method, as studied by Dryja and Widlund We present proofs of convergenceof these iterative methods when applied to the mixed finite element case, and also show that the rate of convergence is independent of the mesh parmeter h. We also present numerical results of tests using these methods with many subdomains and a certain coarse mesh model problem, which improves the rate of convergence. The results indicate a rate of convergence, which is independent of the mesh parameter h and even the number of subdomains. We have also tested the methods on problems in which the discontinuity in the coefficients of the elliptic operator is large. Such large variations in the coefficients occur in certain applications involving flow in porous media. The rate of convergence remains independent of the jump in the discontinuity, but the accuracy of the pressure deteriorates. See the section on numerical results, in Chapter 2.

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