Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient

This paper addresses a multi-scale finite element method for second order linear elliptic equations with rough coefficients, which is based on the compactness of the solution operator, and does not depend on any scale-separation or periodicity assumption of the coefficient. We consider a special type of basis functions, the multi-scale basis, which are harmonic on each element and show that they have optimal approximation property for fixed local boundary conditions. To build the optimal local boundary conditions, we introduce a set of interpolation basis functions, and reduce our problem to approximating the interpolation residual of the solution space on each edge of the coarse mesh. And this is achieved through the singular value decompositions of some local oversampling operators. Rigorous error control can be obtained through thresholding in constructing the basis functions. The optimal interpolation basis functions are also identified and they can be constructed by solving some local least square problems. Numerical results for several problems with rough coefficients and high contrast inclusions are presented to demonstrate the capacity of our method in identifying and exploiting the compact structure of the local solution space to achieve computational savings