Mixed Interior Penalty Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

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

Abstract: This article is concerned with developing efficient discontinuous Galerkin methods for approximating viscosity (and classical) solutions of fully nonlinear second-order elliptic and parabolic partial differential equations (PDEs) including the Monge–Ampère equation and the Hamilton–Jacobi–Bellman equation. A general framework for constructing interior penalty discontinuous Galerkin (IP-DG) methods for these PDEs is presented. The key idea is to introduce multiple discrete Hessians for the viscosity solution as a means to characterize the behavior of the function. The PDE is rewritten in a mixed form composed of a single nonlinear equation paired with a system of linear equations that defines multiple Hessian approximations. To form the single nonlinear equation, the nonlinear PDE operator is replaced by the projection of a numerical operator into the discontinuous Galerkin test space. The numerical operator uses the multiple Hessian approximations to form a numerical moment which fulfills consistency and g-monotonicity requirements of the framework. The numerical moment will be used to design solvers that will be shown to help the IP-DG methods select the “correct” solution that corresponds to the unique viscosity solution. Numerical experiments are also presented to gauge the effectiveness and accuracy of the proposed mixed IP-DG methods.

Additional Information

Numerical Methods for Partial Differential Equations, 30(5), 1538-1557
Language: English
Date: 2014
Fully nonlinear PDEs, viscosity solutions, discontinuous Galerkin methods

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