Daniel Elfverson: A local orthogonal decomposition method for elliptic multiscale problems

We study the numerical approximation of elliptic problems with heterogeneous and highly varying data without any assumptions on scale separation or periodicity. Problems of this type arise in many branches of scientific computing, for example in porous media flow and in composite materials. Standard finite element methods perform poorly for this kind of problems because the data describing the problem needs to be resolved. We propose a local orthogonal decomposition (LOD) method based on a corrected basis calculated on localized patches both for a continuous and discontinuous Galerkin method. The corrected basis functions takes the fine scale variations in the data into account. We prove convergence independently of the variation in the data of the proposed method, both in a symmetric and Petrov-Galerkin formulation. Finally, we will apply this framework for solving the Buckley-Leverett equation and also to elliptic problems where the mesh does not resolve the boundary of the computational domain.