Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients

Type
Article

Authors
Bonito, Andrea
DeVore, Ronald A.
Nochetto, Ricardo H.

KAUST Grant Number
KUS-C1-016-04

Date
2013-01

Abstract
Elliptic PDEs with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electromagnetic field propagation on heterogeneous media, and diffusion processes on rough surfaces. The standard approach to numerically treating such problems using finite element methods is to assume that the discontinuities lie on the boundaries of the cells in the initial triangulation. However, this does not match applications where discontinuities occur on curves, surfaces, or manifolds, and could even be unknown beforehand. One of the obstacles to treating such discontinuity problems is that the usual perturbation theory for elliptic PDEs assumes bounds for the distortion of the coefficients in the L∞ norm and this in turn requires that the discontinuities are matched exactly when the coefficients are approximated. We present a new approach based on distortion of the coefficients in an Lq norm with q < ∞ which therefore does not require the exact matching of the discontinuities. We then use this new distortion theory to formulate new adaptive finite element methods (AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in the sense of distortion versus number of computations, and report insightful numerical results supporting our analysis. © 2013 Societ y for Industrial and Applied Mathematics.

Citation
Bonito A, DeVore RA, Nochetto RH (2013) Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients. SIAM J Numer Anal 51: 3106–3134. Available: http://dx.doi.org/10.1137/130905757.

Acknowledgements
Department of Mathematics, Texas A&M University, College Station, TX 77843 (bonito@math.tamu.edu, rdevore@math.tamu.edu). The first author was partially supported by NSF grant DMS-1254618 and ONR grant N000141110712. The second author was partially supported by ONR grants N00014-12-1-0561 and N00014-11-1-0712, NSF grant DMS-12227151, and award KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST).Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742 (rhn@math.umd.edu). This author was partially supported by NSF grant DMS-1109325.

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Numerical Analysis

DOI
10.1137/130905757

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