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Analysis of an algorithm for generating locally optimal meshes for $L_2$ approximation by discontinuous piecewise polynomials

Authors: Y. Tourigny and M. J. Baines
Journal: Math. Comp. 66 (1997), 623-650
MSC (1991): Primary 41A30; Secondary 65D15
MathSciNet review: 1397447
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Abstract: This paper discusses the problem of constructing a locally optimal mesh for the best $L_2$ approximation of a given function by discontinuous piecewise polynomials. In the one-dimensional case, it is shown that, under certain assumptions on the approximated function, Baines' algorithm [M.J. Baines, Math. Comp., 62 (1994), pp. 645-669] for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh. The rate of convergence is investigated. A two-dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self-adjusting. Numerical results in one and two dimensions are presented.

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Additional Information

Y. Tourigny
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

M. J. Baines
Affiliation: Department of Mathematics, University of Reading, P.O. Box 220, Reading RG6 6AF, United Kingdom

Keywords: $L_2$ approximation, discontinuous piecewise polynomials, adjustable nodes, grid generation, triangulation
Received by editor(s): July 27, 1995
Received by editor(s) in revised form: March 13, 1996
Article copyright: © Copyright 1997 American Mathematical Society