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

Alexandre Caboussat; Dimitrios Gourzoulidis; Marco Picasso

doi:10.5802/crmeca.222

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

Alexandre Caboussat ¹ ; Dimitrios Gourzoulidis ^1,² ; Marco Picasso ²

¹ Geneva School of Business Administration (HEG-GE), University of Applied Sciences and Arts Western Switzerland (HES-SO), 1227 Carouge, Geneva, Switzerland
² Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne (EPFL) 1018 Lausanne, Switzerland

Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 277-292.

Résumé

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.

Reçu le : 2022-12-14
Accepté le : 2023-09-06
Première publication : 2023-10-16
Publié le : 2024-04-26

DOI : 10.5802/crmeca.222

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

Affiliations des auteurs :

Alexandre Caboussat ¹ ; Dimitrios Gourzoulidis ^{1, 2} ; Marco Picasso ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Alexandre Caboussat and Dimitrios Gourzoulidis and Marco Picasso},
     title = {An adaptive least-squares algorithm for the elliptic {Monge{\textendash}Amp\`ere} equation},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {277--292},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {351},
     number = {S1},
     year = {2023},
     doi = {10.5802/crmeca.222},
     language = {en},
}

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%A Dimitrios Gourzoulidis
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Alexandre Caboussat; Dimitrios Gourzoulidis; Marco Picasso. An adaptive least-squares algorithm for the elliptic Monge–Ampère equation. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 277-292. doi : 10.5802/crmeca.222. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.222/

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