Error estimates and adaptivity for a least-squares method applied to the Monge-Amp\`ere equation

TL;DR

This paper develops new a posteriori error estimators and an adaptive mesh refinement strategy for a nonlinear least-squares method solving the Monge-Ampère equation, improving accuracy and efficiency.

Contribution

It introduces novel error indicators and a coupled second-order system formulation for the Monge-Ampère equation, enabling effective adaptive refinement.

Findings

Errors scale appropriately in different norms.

A posteriori indicators effectively guide mesh refinement.

Numerical tests confirm the robustness of the adaptive method.

Abstract

We introduce novel a posteriori error indicators for a nonlinear least-squares solver for smooth solutions of the Monge--Amp\`ere equation on convex polygonal domains in $\mathbb{R}^2$. At each iteration, our iterative scheme decouples the problem into (i) a pointwise nonlinear minimization problem and (ii) a linear biharmonic variational problem. For the latter, we derive an equivalence to a biharmonic problem with Navier boundary conditions and solve it via mixed piecewise-linear finite elements. Reformulating this as a coupled second-order system, we derive a priori and a posteriori $\mathbb{P}^1$ finite element error estimators and we design a robust adaptive mesh refinement strategy. Numerical tests confirm that errors in different norms scale appropriately. Finally, we demonstrate the effectiveness of our a posteriori indicators in guiding mesh refinement.