Least-squares/relaxation method for the numerical solution of a 2D Pucci’s equation

Alexandre Caboussat (Geneva School of Business Administration (Haute Ecole de Gestion de Genève), University of Applied Sciences Western Switzerland (HES-SO), Carouge, Switzerland)

Abstract

The numerical solution of the Dirichlet problem for an elliptic Pucci’s equation in two dimensions of space is addressed by using a least-squares approach. The algorithm relies on an iterative relaxation method that decouples a variational linear elliptic PDE problem from the local nonlinearities. The approximation method relies on mixed low order finite element methods.

The least-squares framework allows to revisit and extend the approach and the results presented in [Caffarelli, Glowinski, 2008] to more general cases. Numerical results show the convergence of the iterative sequence to the exact solution, when such a solution exists. The robustness of the approach is highlighted, when dealing with various types of meshes, domains with curved boundaries, nonconvex domains, or non-smooth solutions.

Keywords

Pucci’s equation, least-squares method, nonlinear constrained minimization, Newton method, mixed finite elements

2010 Mathematics Subject Classification

35F30, 49M15, 49M20, 65K10, 65N30

Full Text (PDF format)

Dedicated to Professor Roland Glowinski on the occasion of his 80th birthday

This work has been supported by the Swiss National Science Foundation (Grant SNF 165785), and the National Science Foundation (Grant NSF DMS-0913982).

Received 30 October 2017

Accepted 30 July 2019

Published 2 April 2020