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| The Multiscale Finite Element Method (MsFEM) is a Finite Element type approximation method for multiscale PDEs, where the basis functions used to generate the approximation space are precomputed and are specifically adapted to the problem at hand. Many ways to define these basis functions have been proposed. Here, we introduce and analyze a specific MsFEM variant, in the spirit of Crouzeix-Raviart elements, where the continuity of the solution accross the mesh edges is enforced only in a weak sense.
Our motivation stems from our wish to address multiscale problems for which implementing flexible boundary conditions on mesh elements is of particular interest. A prototypical situation is that of perforated media, where the accuracy of the numerical solution is generically very sensitive to the choice of values of the basis functions on the boundaries of elements. The Crouzeix-Raviart type elements we construct then provide an advantageous flexibility, as shown by our numerical results.
References: C. Le Bris, F. Legoll and A. Lozinski, MsFEM \`{a} la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems, Chin. Ann. Math. Series B, 34B (2013), 1-26 and arXiv 1307.0876. |
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