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 Partition of Unity Method (PUM). The standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data; e.g., the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point, etc. The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape-functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space. In this way, the local approximability afforded by the special functions is included in the approximation, 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 unity based methods where this is a major issue; (2) the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation (this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods); and (3) linear dependencies in the system of equations are resolved by employing an easy modification of the direct 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.