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Mixed interior penalty discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions
Author(s) -
Feng Xiaobing,
Lewis Thomas
Publication year - 2014
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21856
Subject(s) - mathematics , discontinuous galerkin method , nonlinear system , partial differential equation , hessian matrix , viscosity solution , galerkin method , mathematical analysis , parabolic partial differential equation , numerical analysis , operator (biology) , finite element method , biochemistry , chemistry , physics , repressor , quantum mechanics , gene , transcription factor , thermodynamics
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.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1538–1557, 2014