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Domain decomposition algorithms. (English) Zbl 0809.65112
Iserles, A. (ed.), Acta Numerica 1994. Cambridge: Cambridge University Press. 61-143 (1994).
Summary: Domain decomposition refers to divide and conquer techniques for solving partial differential equations by iteratively solving subproblems defined on smaller subdomains. The principal advantages include enhancement of parallelism and localized treatment of complex and irregular geometries, singularities, and anomalous regions. Additionally, domain decomposition can sometimes reduce the computational complexity of the underlying solution method.
In this article, we survey iterative domain decomposition techniques that have been developed in recent years for solving several kinds of partial differential equations, including elliptic, parabolic, and differential systems such as the Stokes problem and mixed formulations of elliptic problems. We focus on describing the salient features of the algorithms and describe them using easy to understand matrix notation. In the case of elliptic problems, we also provide an introduction to the convergence theory, which requires some knowledge of finite element spaces and elementary functional analysis.
For the entire collection see [Zbl 0797.00003].

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
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