Stable Generalized Finite Element Method (SGFEM)

TL;DR

This paper introduces the Stable GFEM, a modification of the Generalized Finite Element Method that improves the conditioning of the stiffness matrix, making it more robust and accurate for complex problems.

Contribution

The paper proposes the Stable GFEM, which enhances the conditioning and robustness of GFEM, addressing a key numerical stability issue.

Findings

SGFEM has better-conditioned stiffness matrices than GFEM.

SGFEM maintains accuracy despite parameter variations.

Demonstrated robustness on multiple example problems.

Abstract

The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions, which are also known as the enrichments, mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully used to solve a variety of problems with complicated features and microstructure. However, the stiffness matrix of GFEM is badly conditioned (much worse compared to the standard FEM) and there could be a severe loss of accuracy in the computed solution of the associated linear system. In this paper, we address this issue and propose a modification of the GFEM, referred to as the Stable GFEM (SGFEM). We show that the conditioning of the stiffness matrix of SGFEM is not worse than that of the standard FEM. Moreover, SGFEM is very robust with respect to the parameters of the enrichments. We show these features of SGFEM on several examples.