On the Application of Stable Generalized Finite Element Method for Quasilinear Elliptic Two-Point BVP

TL;DR

This paper explores the use of the stable generalized finite element method (SGFEM) to accurately solve one-dimensional quasilinear elliptic equations with multiple interfaces, overcoming issues caused by discontinuities and improving local conservation properties.

Contribution

It introduces a stable GFEM approach with enrichment functions for quasilinear elliptic problems with discontinuities, ensuring optimal convergence and improved local conservation.

Findings

SGFEM maintains optimal convergence rates.

Numerical examples demonstrate improved accuracy with discontinuities.

Lagrange multiplier technique enforces local conservation effectively.

Abstract

In this paper, we discuss the application of the Generalized Finite Element Method (GFEM) to approximate the solutions of quasilinear elliptic equations with multiple interfaces in one dimensional space. The problem is characterized by spatial discontinuity of the elliptic coefficient that depends on the unknown solution. It is known that unless the partition of the domain matches the discontinuity configuration, accuracy of standard finite element techniques significantly deteriorates and standard refinement of the partition may not suffice. The GFEM is a viable alternative to overcome this predicament. It is based on the construction of certain enrichment functions supplied to the standard space that capture effects of the discontinuity. This approach is called stable (SGFEM) if it maintains an optimal rate of convergence and the conditioning of GFEM is not worse than that of the standard FEM. A convergence analysis is derived and performance of the method is illustrated by several numerical examples. Furthermore, it is known that typical global formulations such as FEMs do not enjoy the numerical local conservation property that is crucial in many conservation law-based applications. To remedy this issue, a Lagrange multiplier technique is adopted to enforce the local conservation. A numerical example is given to demonstrate the performance of proposed technique.