C Penalty Methods for the Fully Nonlinear Monge-ampère Equation
نویسندگان
چکیده
In this paper, we develop and analyze C0 penalty methods for the fully nonlinear Monge-Ampère equation det(D2u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.
منابع مشابه
Pseudo Time Continuation and Time Marching Methods for Monge-ampère Type Equations
We discuss the performance of three numerical methods for the fully nonlinear Monge-Ampère equation. The first two are pseudo time continuation methods while the third is a pure pseudo time marching algorithm. The pseudo time continuation methods are shown to converge for smooth data on a uniformly convex domain. We give numerical evidence that they perform well for the nondegenerate Monge-Ampè...
متن کاملC0 penalty methods for the fully nonlinear Monge-Ampère equation
In this paper, we develop and analyze C0 penalty methods for the fully nonlinear Monge-Ampère equation det(D2u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well a...
متن کاملQuasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampère systems
This paper studies the pressureless Euler-Poisson system and its fully non-linear counterpart, the Euler-Monge-Ampère system, where the fully non-linear Monge-Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of bo...
متن کاملWide Stencil Finite Difference Schemes for the Elliptic Monge-ampère Equation and Functions of the Eigenvalues of the Hessian
Certain fully nonlinear elliptic Partial Differential Equations can be written as functions of the eigenvalues of the Hessian. These include: the Monge-Ampère equation, Pucci’s Maximal and Minimal equations, and the equation for the convex envelope. In this article we build convergent monotone finite difference schemes for the aforementioned equations. Numerical results are presented.
متن کاملComparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampère type∗
We study fully nonlinear partial differential equations of Monge-Ampère type involving the derivates with respect to a family X of vector fields. The main result is a comparison principle among viscosity subsolutions, convex with respect to X , and viscosity supersolutions (in a weaker sense than usual), which implies the uniqueness of solution to the Dirichlet problem. Its assumptions include ...
متن کامل