We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Ampére type. These surfaces need not be continuous up to the boundary of the domain and the Dirichlet boundary condition must be interpreted in a weak sense. As a consequence, sub-solutions do not always lie below super-solutions, standard comparison principles fail, and existing conver-gence theorems break down. By relying on a geometric interpretation of weak solutions, we prove a relaxed comparison principle that applies only in the in-terior of the domain. We provide a general framework for proving existence and stability results for consistent, monotone finite difference approximations and modify the Barles-Souganidis convergence framework to show convergence in the interior of the domain. We describe a convergent scheme for the pres-cribed Gaussian curvature equation and present several challenging examples to validate these results.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Elliptic partial differential equations
- Finite difference methods.
- Gaussian curvature
- Monge-Ampére equations
- Viscosity solutions