On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators

TL;DR

This paper introduces a finite difference method for approximating the principal eigenvalue and eigenfunction of various nonlinear elliptic operators, with proven convergence and demonstrated effectiveness through examples.

Contribution

It develops a novel finite difference approach that handles a broad class of nonlinear elliptic operators, including nondivergence form and fully nonlinear types, with theoretical convergence guarantees.

Findings

Method accurately computes eigenvalues in tested examples

Convergence of the method is rigorously proven

Effective for both linear and nonlinear elliptic operators

Abstract

We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators. The principal eigenvalue is computed by solving a finite-dimensional nonlinear min-max optimization problem. We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method.