Finite element solution of a radiation/propagation problem for a Helmholtz equation with a compactly supported nonlinearity

TL;DR

This paper develops and analyzes a finite element method for solving a nonlinear Helmholtz equation modeling electromagnetic response of objects, including error estimates and well-posedness considerations.

Contribution

It introduces a finite element approach for nonlinear Helmholtz equations with rigorous analysis of stability and error bounds, extending to three-dimensional cases.

Findings

Finite element method with Courant-type elements is effective for the nonlinear Helmholtz problem.

The method satisfies a discrete inf-sup condition ensuring stability.

A quasi-optimal error estimate is established under certain assumptions.

Abstract

A finite element approach for approximating the solution of a mathematical model for the response of a penetrable, bounded object (obstacle) to the excitation by an external electromagnetic field is presented and investigated. The model consists of a nonlinear Helmholtz equation that is reduced to a spherical domain. As a specific example, we consider a finite element method consisting of Courant-type elements with curved edges at the boundary of a circular computational domain in the two-dimensional case. We examine this method and more general conforming methods -- including three-dimensional ones -- with comparable properties for their well-posedness; in particular, the validity of a discrete inf-sup condition of the modified sesquilinear form uniformly with respect to both the truncation and the mesh parameters is shown. Under suitable assumptions to the nonlinearities, a quasi-optimal error estimate is obtained. Finally, the satisfiability of the approximation property of the finite element space required for the solvability of a class of adjoint linear problems is discussed.