53 research outputs found
Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions
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 C0 or C1 functions. Standard discretizations of the
type considered in this paper have been previous 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-Amp\`ere equation
Iterative methods for k-Hessian equations
On a domain of the n-dimensional Euclidean space, and for an integer
k=1,...,n, the k-Hessian equations are fully nonlinear elliptic equations for k
>1 and consist of the Poisson equation for k=1 and the Monge-Ampere equation
for k=n. We analyze for smooth non degenerate solutions a 9-point finite
difference scheme. We prove that the discrete scheme has a locally unique
solution with a quadratic convergence rate. In addition we propose new
iterative methods which are numerically shown to work for non smooth solutions.
A connection of the latter with a popular Gauss-Seidel method for the
Monge-Ampere equation is established and new Gauss-Seidel type iterative
methods for 2-Hessian equations are introduced
Pseudo transient continuation and time marching methods for Monge-Ampere type equations
We present two numerical methods for the fully nonlinear elliptic
Monge-Ampere equation. The first is a pseudo transient continuation method and
the second is a pure pseudo time marching method. The methods are proven to
converge to a strictly convex solution of a natural discrete variational
formulation with conforming approximations. The assumption of existence
of a strictly convex solution to the discrete problem is proven for smooth
solutions of the continuous problem and supported by numerical evidence for non
smooth solutions
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