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arXiv · 2507.17569

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

Alexandre Caboussat·Anna Peruso·Marco Picasso
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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.

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Alexandre Caboussat, Anna Peruso, Marco Picasso. 2025-07-23. Error estimates and adaptivity for a least-squares method applied to the Monge-Amp\`ere equation. https://arxiv.org/abs/2507.17569

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