Coarse spaces over the ages

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0911.5725.pdf (378.5 KB)

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https://arxiv.org/abs/0911.5725

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UMBC Mathematics and Statistics Department

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https://orcid.org/0000-0002-8053-8956

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2010

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conference papers and proceedings postprints
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Abstract

The objective of this paper is to explain the principles of the design of a coarse space in a simplified way and by pictures. The focus is on ideas rather than on a more historically complete presentation. That can be found, e.g., in Widlund [2008]. Also, space limitation does not allow us to even the mention many important methods and papers that should be rightfully included. The coarse space facilitates a global exchange of information in multigrid and domain decomposition methods for elliptic problems. This exchange is necessary, because the solution is non-local: its value at any point depends on the right-hand-side at any other point. Both multigrid and domain decomposition combine a global correction in coarse space with local corrections, called smoothing in multigrid and subdomain solves in domain decomposition. In multigrid the coarse space is large (typically, the mesh ratio is 2 or 3 at most) and the local solvers are not very powerful (usually, relaxation). In domain decomposition, the coarse space is small (just one or a few degrees of freedom per subdomain), and the local solvers are powerful (direct solvers on subdomain). But the mathematics is more or less the same.