Abstract
In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.
Highlights
The Monge–Ampère equation can be considered as the prototypical example of fully nonlinear elliptic equations [11,21,29]
The numerical solution of the 3D elliptic Monge–Ampère equation has been discussed in [8] using piecewise P3 continuous finite element approximations
To the best of our knowledge, the method discussed in the present article is one of the very few able to solve the 3D elliptic Monge–Ampère equation on domains with curved boundaries, using piecewise P1 continuous finite element approximations associated with unstructured meshes, while preserving optimal, or nearly optimal, orders of convergence for the approximation errors, including situations where the solution does not have the C2(Ω ) regularity
Summary
The Monge–Ampère equation can be considered as the prototypical example of fully nonlinear elliptic equations [11,21,29]. Using a relaxation algorithm to minimize such a distance, we obtained a solution method where one solves alternatively, until convergence, a sequence of linear variational problems (to be approximated by mixed finite element methods) and a sequence of cubicly constrained algebraic optimization problems. The numerical solution of the 3D elliptic Monge–Ampère equation has been discussed in [8] using piecewise P3 continuous finite element approximations. To the best of our knowledge, the method discussed in the present article is one of the very few able to solve the 3D elliptic Monge–Ampère equation on domains with curved boundaries, using piecewise P1 continuous finite element approximations associated with unstructured meshes, while preserving optimal, or nearly optimal, orders of convergence for the approximation errors, including situations where the solution does not have the C2(Ω ) regularity
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