Multiscale mixed finite elements
For researchers solving multiscale elliptic PDEs, this provides a novel method that is efficient and provably convergent without structural assumptions.
The paper proposes a mixed finite element method for elliptic multiscale problems using localized orthogonal decomposition, achieving high approximation properties with cheap, parallelizable local computations. Numerical experiments demonstrate its applicability and convergence independent of scale separation.
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.