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The generalized finite element method: An example of its implementation and illustration of its performance. (English) Zbl 0955.65080

Summary: The generalized finite element method (GFEM) was introduced by the authors [Comput. Methods Appl. Mech. Eng. 181, No. 1-3, 43-69 (2000)] as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect 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 finite 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 affect 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.


MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65Y20Complexity and performance of numerical algorithms
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation