The Carleman contraction mapping method for quasilinear elliptic equations with over-determined boundary data

TL;DR

This paper introduces a globally convergent numerical method combining Carleman estimates and quasi-reversibility to solve quasilinear elliptic PDEs with over-determined boundary data, demonstrating robustness and convergence.

Contribution

The paper develops a new Carleman contraction mapping method that guarantees global convergence for quasilinear elliptic equations with noisy boundary data.

Findings

Method converges reliably even with noisy data

Numerical examples validate the approach

Provides a new framework for solving over-determined boundary problems

Abstract

We propose a globally convergent numerical method to compute solutions to a general class of quasi-linear PDEs with both Neumann and Dirichlet boundary conditions. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the PDE under consideration. To find this fixed point, we define a recursive sequence with an arbitrary initial term using the same manner as in the proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution. On the other hand, we also show that our method delivers reliable solutions even when the given data are noisy. Numerical examples are presented.