A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications

TL;DR

This paper introduces a novel iterative numerical method leveraging Carleman estimates to efficiently solve over-determined boundary value problems for quasilinear elliptic PDEs, with proven exponential convergence and practical applications.

Contribution

The paper develops a new Carleman-based iterative scheme for quasilinear elliptic equations with over-determined data, ensuring fast convergence without requiring a good initial guess.

Findings

Method converges exponentially fast.

Applicable to general quasilinear elliptic and Hamilton-Jacobi equations.

Numerical experiments demonstrate effectiveness.

Abstract

We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented.