p-version of the generalized FEM using mesh-based handbooks with applications to multiscale problems

In this paper, we analyse the p-convergence of a new version of the generalized finite element method (generalized FEM or GFEM) which employs mesh-based handbook functions which are solutions of boundary value problems in domains extracted from vertex patches of the employed mesh and are pasted into the global approximation by the partition of unity method (PUM). We show that the p-version of our GFEM is capable of achieving very high accuracy for multiscale problems which may be impossible to solve using the standard FEM. We analyse the effect of the main factors affecting the accuracy of the method namely: (a) The data and the buffer included in the handbook domains, and (b) The accuracy of the numerical construction of the handbook functions. We illustrate the robustness of the method by employing as model problem the Laplacian in a domain with a large number of closely spaced voids. Similar robustness can be expected for problems of heat-conduction and elasticity set in domains with a large number of closely spaced voids, cracks, inclusions, etc