Abstract
We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge–Ampere equation using finite dimensional spaces of piecewise polynomial C 1 functions. Discretizations of the type considered in this paper have been previously analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge–Ampere equation.
Highlights
In this paper we prove a convergence result for the numerical approximation of solutions to the Dirichlet problem for the Monge-Ampere equation det D2u = f in Ω, u = g on ∂Ω, by elements of a space Vh of piecewise polynomial functions of some degree k ≥ 2 which are either globally C0 or globally C1
The convergence result of this paper addresses this issue
We present a theory which explains why standard discretizations of the type considered in this paper do converge for non smooth solutions of the MongeAmpere equation
Summary
In this paper we prove a convergence result for the numerical approximation of solutions to the Dirichlet problem for the Monge-Ampere equation (1.1). In [4], we analyzed the discretization (1.3) for both C0 and C1 approximations and gave numerical evidence of convergence for non smooth solutions if one uses Lagrange elements and a time marching method. The latter approach is less general in the sense that it does not apply to collocation type discretizations such as the standard finite difference method It is a standard technique in the analysis of Aleksandrov solutions of the Monge-Ampere equation, e.g. The approach of this paper can be adapted to explain the numerical results with singular data presented in [11] For another example, again for a smooth uniformly convex domain, one can use isogeometric analysis as in [1] for Ω = Ω.
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