The generalized finite element method: an example of its implementation and illustration of its performance

The generalized nite element method (GFEM) was introduced in Reference [1] as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial nite element spaces are augmented by adding special functions which re ect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing nite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not a ect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM.