Numerical solution of boundary value problems for the eikonal equation in an anisotropic medium

TL;DR

This paper develops a numerical method for solving boundary value problems of the eikonal equation in anisotropic media, using monotone approximations and finite-element methods, with demonstrated effectiveness through 2D examples.

Contribution

It introduces a numerical approach for anisotropic eikonal equations based on monotone approximations of a related diffusion-reaction problem.

Findings

Effective numerical solutions for 2D anisotropic eikonal problems.

Validation of the method through numerical examples.

Use of finite-element approximation in computations.

Abstract

A Dirichlet problem is considered for the eikonal equation in an anisotropic medium. The nonlinear boundary value problem (BVP) formulated in the present work is the limit of the diffusion-reaction problem with a reaction parameter tending to infinity. To solve numerically the singularly perturbed diffusion-reaction problem, monotone approximations are employed. Numerical examples are presented for a two-dimensional BVP for the eikonal equation in an anisotropic medium. The standard piecewise-linear finite-element approximation in space is used in computations.