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A least-squares method for the numerical solution of theDirichlet problem for the elliptic monge − ampère equation in dimension two∗

Published online by Cambridge University Press:  03 June 2013

Alexandre Caboussat ,
Roland Glowinski  and
Danny C. Sorensen
Alexandre Caboussat
Affiliation:
Haute École de Gestion/Geneva School of Business Administration, Genève, Switzerland. [email protected] University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, 77204-3008 Texas, USA; [email protected]
Roland Glowinski
Affiliation:
University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, 77204-3008 Texas, USA; [email protected]
Danny C. Sorensen
Affiliation:
Rice University, Department of Computational and Applied Mathematics, MS 134, Houston, 77251-1892 Texas, USA; [email protected]
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Abstract

We address in this article the computation of the convex solutions of the Dirichletproblem for the real elliptic Monge − Ampère equation for general convex domains in twodimensions. The method we discuss combines a least-squares formulation with a relaxationmethod. This approach leads to a sequence of Poisson − Dirichlet problems and anothersequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finiteelement approximations with a smoothing procedure are used for the computer implementationof our least-squares/relaxation methodology. Domains with curved boundaries are easilyaccommodated. Numerical experiments show the convergence of the computed solutions totheir continuous counterparts when such solutions exist. On the other hand, when classicalsolutions do not exist, our methodology produces solutions in a least-squares sense.

Keywords

Monge − Ampère equationleast-squares methodbiharmonic problemconjugate gradient methodquadratic constraint minimizationmixed finite element methods
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 780 - 810
Copyright
© EDP Sciences, SMAI, 2013

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