An adaptive least-squares algorithm for the elliptic Monge–Ampère equation

Abstract

We address the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities of the equation, within a splitting approach. The approximation relies on mixed low order finite element methods with regularization techniques. In order to account for data singularities in non-smooth cases, we introduce an adaptive mesh refinement technique. The error indicator is based an independent formulation of the Monge–Ampère equation under divergence form, which allows to explicit a residual term. We show that the error is bounded from above by an a posteriori error indicator plus an extra term that remains to be estimated. This indicator is then used within the existing least-squares framework. The results of numerical experiments support the convergence of our relaxation method to a convex classical solution, if such a solution exists. Otherwise they support convergence to a generalized solution in a least-squares sense. Adaptive mesh refinement proves to be efficient, robust, and accurate to tackle test cases with singularities.

Keywords

• Fully nonlinear PDE
• Monge–Ampère equation
• Least-squares algorithm
• Mixed finite elements
• Adaptive mesh refinement
• Non-smooth problems