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Published October 2010 | Published
Journal Article Open

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


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 coefficientdependent 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 O(h^(1/2−ε)) 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.

Additional Information

© 2010 American Mathematical Society. Received by the editor February 24, 2009. Article electronically published on May 25, 2010. 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.

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August 22, 2023
August 22, 2023