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A new multiscale finite element method for high-contrast elliptic interface problems

Authors: C.-C. Chu, I. G. Graham and T.-Y. Hou
Journal: Math. Comp. 79 (2010), 1915-1955
MSC (2010): Primary 65N12, 65N30
Published electronically: May 25, 2010
MathSciNet review: 2684351
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Abstract: We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of flow in a porous medium containing a number of inclusions of low (or high) permeability embedded in a matrix of high (respectively low) permeability. Our method is $H^1$- conforming, with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise. A key point is the introduction of novel coefficient-dependent boundary conditions for the subgrid problems. Under moderate assumptions, we prove that our methods have (optimal) convergence rate of $O(h)$ in the energy norm and $O(h^2)$ in the $L_2$ norm where $h$ is the (coarse) mesh diameter and the hidden constants in these estimates are independent of the “contrast” (i.e. ratio of largest to smallest value) of the PDE coefficient. For standard elements the best estimate in the energy norm would be $\mathcal {O}(h^{1/2-\varepsilon })$ with a hidden constant which in general depends on the contrast. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges.

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Additional Information

C.-C. Chu
Affiliation: Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, California 91125
Address at time of publication: Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712

I. G. Graham
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
MR Author ID: 76020

T.-Y. Hou
Affiliation: Department of Applied and Computational Mathematics, California Insitute of Technology, Pasadena, California 91125

Keywords: Second-order elliptic problems, interfaces, high contrast, multiscale finite elements, non-periodic media, convergence.
Received by editor(s): February 24, 2009
Published electronically: May 25, 2010
Additional Notes: The authors thank Rob Scheichl and Jens Markus Melenk for useful discussions. The second author acknowledges financial support from the Applied and Computational Mathematics Group at California Institute of Technology. The research of the third author was supported in part by an NSF Grant DMS-0713670 and a DOE Grant DE-FG02-06ER25727.
Article copyright: © Copyright 2010 American Mathematical Society