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

The generalized finite element method (GFEM) was introduced in Reference [1 ] as a combination of the standard FEM and the partition of unity method. T he standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary v alue problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the pa rtition of unity corresponding to the standard linear vertex shape function s and are pasted to the existing finite element basis to construct a confor ming approximation. The essential boundary conditions can be imposed exactl y as in the standard FEM. Adaptive numerical quadrature is used to ensure t hat the errors in integration do not affect the accuracy of the approximati on. This paper gives an example of how the GFEM can be developed for the La placian in domains with multiple elliptical voids and illustrates implement ation issues and the superior accuracy of the GFEM versus the standard FEM. Copyright (C) 2000 John Wiley & Sons, Ltd.