Stable Generalized Finite Element Method and associated iterative schemes; application to interface problems

TL;DR

This paper introduces a stable version of the Generalized Finite Element Method (GFEM) that maintains good conditioning through an angle condition and develops an efficient iterative solver, improving computational efficiency for interface problems.

Contribution

It proposes a Stable GFEM (SGFEM) with a bounded angle condition ensuring robustness and introduces a robust iterative solver that significantly reduces computational time.

Findings

SGFEM maintains conditioning comparable to standard FEM.

The angle condition ensures stability across mesh refinements.

The iterative solver reduces wall-clock time significantly.

Abstract

The Generalized Finite Element Method (GFEM) is an extension of the Finite Element Method (FEM), where the standard finite element space is augmented with a space of non-polynomial functions, called the enrichment space. The functions in the enrichment space mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully applied to a wide range of problems. However, it often suffers from bad conditioning, i.e., its conditioning may not be robust with respect to the mesh and in fact, the conditioning could be much worse than that of the standard FEM. In this paper, we present a numerical study that shows that if the "angle" between the finite element space and the enrichment space is bounded away from 0, uniformly with respect to the mesh, then the GFEM is stable, i.e., the conditioning of GFEM is not worse than that of the standard FEM. A GFEM with this property is called a Stable GFEM (SGFEM). The last part of the paper is devoted to the derivation of a robust iterative solver exploiting this angle condition. It is shown that the required "wall-clock" time is greatly reduced compared to popular GFEMs used in the literature.