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Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes (1994)

Chan, Tony F. ; Zou, Jun

Springer

Numerical algorithms 8 (1994), S. 329-346

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ISSN: 1572-9265

Keywords: Unstructured meshes ; non-nested coarse meshes ; additive Schwarz algorithm ; optimal convergence rate ; 65N30 ; 65F10

Source: Springer Online Journal Archives 1860-2000

Topics: Computer Science , Mathematics

Notes: Abstract We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasi-uniform. In this general setting, our algorithms have the same optimal convergence rate of the usual domain decomposition methods on structured meshes. The condition numbers of the preconditioned systems depend only on the (possibly small) overlap of the substructures and the size of the coares grid, but is independent of the sizes of the subdomains.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF02142697

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Low-rank revealing QR factorizations (1994)

Chan, Tony F. ; Hansen, Per Christian

New York, NY [u.a.] : Wiley-Blackwell

Numerical Linear Algebra with Applications 1 (1994), S. 33-44

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ISSN: 1070-5325

Keywords: Rank revealing QR factorization ; Column pivoting ; Numerical rank ; Subset selection ; Engineering ; Engineering General

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Mathematics

Notes: Rank revealing factorizations are used extensively in signal processing in connection with, for example, linear prediction and signal subspace algorithms. We present an algorithm for computing rank revealing QR factorizations of low-rank matrices. The algorithm produces tight upper and lower bounds for all the largest singular values, thus making it particularly useful for treating rank deficient problems by means of subset selection, truncated QR, etc. The algorithm is similar in spirit to an algorithm suggested earlier by Chan for matrices with a small nullity, and it can also be considered as an extension of ordinary column pivoting.

Additional Material: 1 Ill.