Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces |
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Authors: | BARRETT, JOHN W. ELLIOTT, CHARLES M. |
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Affiliation: | Department of Mathematics, Imperial College London SW7 2BZ Centre for Applied Mathematics, Purdue University West Lafayette, Indiana 47907 |
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Abstract: | This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rn (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H1 and L2 norms overthe approximations l4 of l where l 2. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H1 norm over l4 and in the L2norm over any interior domain l* satisfying l* l** l4 for somedomain l**. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH |
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