Inverted finite elements approximation of the Neumann problem for second order elliptic equations in exterior two-dimensional domains

TL;DR

This paper introduces an inverted finite elements method for solving second order elliptic equations with Neumann boundary conditions in exterior 2D domains, demonstrating convergence and efficiency through analysis and numerical tests.

Contribution

The paper develops a novel inverted finite elements approach tailored for exterior elliptic problems with unbounded coefficients, including convergence analysis and implementation details.

Findings

Method converges in numerical experiments.

High efficiency demonstrated in tests.

Applicable to elliptic equations with coefficients tending to infinity.

Abstract

We use inverted finite elements method for approximating solutions of second order elliptic equations with non-constant coefficients varying to infinity in the exterior of a 2D bounded obstacle, when a Neumann boundary condition is considered. After proposing an appropriate functional framework for the deployment of the method, we analyse its convergence and detail its implementation. Numerical tests performed after implementation confirm convergence and high efficiency of the method.