Meshfree finite difference methods for fully nonlinear elliptic equations

Applied and Computational Mathematics Seminar
Monday, March 7, 2016 - 14:00
1 hour (actually 50 minutes)
Skiles 005
New Jersey Institute of Technology
 The relatively recent introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations.  Convergent, wide-stencil finite difference methods now exist for a variety of problems.  However, these schemes are defined only on uniform Cartesian meshes over a rectangular domain.  We describe a framework for constructing convergent meshfree finite difference approximations for a class of nonlinear elliptic operators.  These approximations are defined on unstructured point clouds, which allows for computation on non-uniform meshes and complicated geometries.  Because the schemes are monotone, they fit within the Barles-Souganidis convergence framework and can serve as a foundation for higher-order filtered methods.  We present computational results for several examples including problems posed on random point clouds, computation of convex envelopes, obstacle problems, Monge-Ampere equations, and non-continuous solutions of the prescribed Gaussian curvature equation.