An Accelerated Method for Nonlinear Elliptic PDE

Abstract

We propose two numerical methods for accelerating the convergence of the standard fixed point method associated with a nonlinear and/or degenerate elliptic partial differential equation. The first method is linearly stable, while the second is provably convergent in the viscosity solution sense. In practice, the methods converge at a nearly linear complexity in terms of the number of iterations required for convergence. The methods are easy to implement and do not require the construction or approximation of the Jacobian. Numerical examples are shown for Bellman’s equation, Isaacs’ equation, Pucci’s equations, the Monge–Ampère equation, a variant of the infinity Laplacian, and a system of nonlinear equations.