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

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.