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Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

Authors:

Thomas LewisAuthors Info & Claims

Journal of Scientific Computing, Volume 77, Issue 3

Pages 1534 - 1565

https://doi.org/10.1007/s10915-018-0765-z

Published: 01 December 2018 Publication History

Abstract

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.

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Digital Library

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Cited By

Nguyen NPeraire J(2024)Hybridizable Discontinuous Galerkin Methods for the Two-Dimensional Monge–Ampère EquationJournal of Scientific Computing10.1007/s10915-024-02604-3100:2Online publication date: 27-Jun-2024
https://dl.acm.org/doi/10.1007/s10915-024-02604-3
Zhong XQiu W(2024)Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEsJournal of Scientific Computing10.1007/s10915-024-02544-y99:3Online publication date: 2-May-2024
https://dl.acm.org/doi/10.1007/s10915-024-02544-y
Chen HFeng XZhang Z(2021)A recovery-based linear C0 finite element method for a fourth-order singularly perturbed Monge-Ampère equationAdvances in Computational Mathematics10.1007/s10444-021-09847-w47:2Online publication date: 1-Apr-2021
https://dl.acm.org/doi/10.1007/s10444-021-09847-w

Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
    2. Numerical analysis
2. Theory of computation

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Published In

cover image Journal of Scientific Computing

Journal of Scientific Computing Volume 77, Issue 3

December 2018

678 pages

ISSN:0885-7474

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Copyright © Copyright © 2018 Springer Science+Business Media, LLC, part of Springer Nature.

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Plenum Press

United States

Publication History

Published: 01 December 2018

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Cited By

Nguyen NPeraire J(2024)Hybridizable Discontinuous Galerkin Methods for the Two-Dimensional Monge–Ampère EquationJournal of Scientific Computing10.1007/s10915-024-02604-3100:2Online publication date: 27-Jun-2024
https://dl.acm.org/doi/10.1007/s10915-024-02604-3
Zhong XQiu W(2024)Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEsJournal of Scientific Computing10.1007/s10915-024-02544-y99:3Online publication date: 2-May-2024
https://dl.acm.org/doi/10.1007/s10915-024-02544-y
Chen HFeng XZhang Z(2021)A recovery-based linear C0 finite element method for a fourth-order singularly perturbed Monge-Ampère equationAdvances in Computational Mathematics10.1007/s10444-021-09847-w47:2Online publication date: 1-Apr-2021
https://dl.acm.org/doi/10.1007/s10444-021-09847-w

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