An Adaptive Local (AL) Basis for Elliptic Problems with Complicated Discontinuous Coefficients

We develop a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. In [5] a semidiscrete method has been introduced to set up an adaptive local finite element basis (AL basis) on a coarse mesh with mesh size H which, typically, does not resolve the matrix of the media while the textbook finite element convergence rates are preserved. This method requires O(log(1/H)^d+1) basis functions per mesh point where d denotes the spatial dimension of the computational domain. We present a fully discrete version of this method, where the AL basis is constructed by solving finite-dimensional localized problems, and which preserves the optimal convergence rates. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)