# American Institute of Mathematical Sciences

October  2016, 9(5): 1269-1298. doi: 10.3934/dcdss.2016051

## Multiscale mixed finite elements

 1 Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden 2 Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden 3 Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Göteborg, Sweden

Received  January 2015 Revised  August 2015 Published  October 2016

In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart--Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.
Citation: Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051
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