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Applied and Computational Math Seminar

Generalized finite difference methods for fully nonlinear elliptic equations 

Brittany Hamfeldt, NJIT 

Location:  Hill 425
Date & time: Friday, 20 September 2019 at 12:00PM - 1:00PM

Abstract: The introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations. We describe a framework for constructing convergent generalised finite difference approximations for a large 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, examples incorporating automatic mesh adaptation, non-continuous surfaces of prescribed Gaussian curvature, Monge-Ampere equations arising in optimal transportation, and Monge-Ampere type equations on the sphere.

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