Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient
For computational scientists solving elliptic PDEs with rough coefficients, this method provides a rigorous, assumption-free approach to multi-scale finite elements with optimal local approximation.
This paper develops a local oversampling method for constructing multi-scale basis functions that achieve optimal local approximation for linear elliptic equations with arbitrarily rough coefficients, without requiring scale-separation or periodicity. Numerical results demonstrate computational savings by exploiting the compact structure of the local solution space.
This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local approximation property. Our methodology is based on the compactness of the solution operator restricted on local regions of the spatial domain, and does not depend on any scale-separation or periodicity assumption of the coefficient. We focus on a special type of basis functions that are harmonic on each element and have optimal approximation property. We first reduce our problem to approximating the trace of the solution space on each edge of the underlying mesh, and then achieve this goal through the singular value decomposition of an oversampling operator. Rigorous error estimates can be obtained through thresholding in constructing the basis functions. Numerical results for several problems with multiple spatial scales and high contrast inclusions are presented to demonstrate the compactness of the local solution space and the capacity of our method in identifying and exploiting this compact structure to achieve computational savings.