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

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.

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Cited by in corpus (4)

• The Carleman convexification method for Hamilton-Jacobi equations on the whole space
• The Carleman contraction mapping method for quasilinear elliptic equations with over-determined boundary data
• Determining initial conditions for nonlinear hyperbolic equations with time dimensional reduction and the Carleman contraction
• A global approach for the inverse scattering problem using a Carleman contraction map