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

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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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Authors and Affiliations

  1. Department of Mathematics, The University of Tennessee, Knoxville, TN, 37996, USA

    Xiaobing Feng

  2. Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC, 27410, USA

    Thomas Lewis

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  1. Xiaobing Feng
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  2. Thomas Lewis
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Correspondence to Xiaobing Feng.

Additional information

This paper is dedicated to Professor Bernardo Cockburn on the occasion of his sixtieth birthday.

The work of both authors was partially supported by the NSF Grant DMS-1620168.

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Feng, X., Lewis, T. Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions. J Sci Comput 77, 1534–1565 (2018). https://doi.org/10.1007/s10915-018-0765-z

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

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