Narrow-stencil finite difference methods for linear second order elliptic problems of non-divergence form

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Kellie Marie Ward (Creator)
Institution
The University of North Carolina at Greensboro (UNCG )
Web Site: http://library.uncg.edu/
Advisor
Thomas Lewis

Abstract: This thesis presents a class of narrow-stencil finite difference methods for approximating the viscosity solution of second-order linear elliptic Dirichlet boundary value problems. The methods are simple to motivate and implement. This thesis proves admissibility and stability results for the simple narrow-stencil finite difference methods as well as optimal convergence rates when the underlying solution to the partial differential equation (PDE) is sufficiently smooth. The results in this thesis extend the analytic techniques first developed by Feng and Lewis when approximating viscosity solutions of fully nonlinear elliptic PDEs using the Lax-Friedrich’s-like method. Numerical tests are presented to gauge the performance of the methods and to validate the convergence results of the thesis.

Additional Information

Publication
Thesis
Language: English
Date: 2021
Keywords
Finite Difference methods, Hamilton-Jacobi-Bellman equations, Narrow-Stencil, Non-Divergence Form, Numerical moment, Viscosity
Subjects
Finite differences
Viscosity solutions
Dirichlet problem
Differential equations, Elliptic

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