The design and analysis of the Generalized Finite Element Method

In this paper, we introduce the Generalized Finite Element Method (GFEM) as a combination of the classical Finite Element Method (FEM) and the Partiti on of Unity Method (PUM. The standard finite element spaces are augmented b y adding special functions which reflect the known information about the bo undary value problem and the input data, e.g., the singular functions obtai ned from the local asymptotic expansion of the exact solution in the neighb orhood of a corner point, etc. The special functions are multiplied with th e partition of unity corresponding to the standard linear vertex shape-func tions and pasted together with the existing finite element basis to constru ct an augmented conforming finite element space. In this way, the local app roximability afforded by the special functions is included in the approxima tion, while maintaining the existing infrastructure of finite element codes . The major features of the GFEM are: (1) the essential boundary conditions can be imposed exactly as in the standard FEM, unlike other partition of u nity based methods where this is a major issue; (2) the accuracy of the num erical integration of the entries of the stiffness matrix and load vector i s controlled adaptively so that the errors in integration of the special fu nctions do not affect the accuracy of the constructed approximation (this i ssue also has not been sufficiently addressed in other implementations of p artition of unity or meshless methods): and (3) linear dependencies in the system of equations are resolved by employing an easy modification of the d irect linear solver. The power of the GFEM for solving problems in domains with complex geometry with less error and less computer resources than the standard FEM is illustrated by numerical examples. (C) 2000 Published by El sevier Science S.A. All rights reserved.