Volume 32, Issue 2
Mixed Interior Penalty Discontinuous Galerkin Methods for One-Dimensional Fully Nonlinear Second Order Elliptic and Parabolic Equations

Xiaobing Feng & Thomas Lewis

J. Comp. Math., 32 (2014), pp. 107-135.

Published online: 2014-04

[An open-access article; the PDF is free to any online user.]

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  • Abstract

This paper is concerned with developing accurate and ecient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin (IP-DG) methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative u_{xx} of the solution u, three independent functions p_1; p_2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative u_{xx}. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity (called g-monotonicity) properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG framework. The g-monotonicity gives the DG methods the ability to select the mathematically "correct" solution (i.e., the viscosity solution) among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.

  • Keywords

Fully nonlinear PDEs Viscosity solutions Discontinuous Galerkin methods

  • AMS Subject Headings

65N30 65M60 35J60 35K55.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-32-107, author = {Xiaobing Feng and Thomas Lewis}, title = {Mixed Interior Penalty Discontinuous Galerkin Methods for One-Dimensional Fully Nonlinear Second Order Elliptic and Parabolic Equations}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {2}, pages = {107--135}, abstract = {

This paper is concerned with developing accurate and ecient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin (IP-DG) methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative u_{xx} of the solution u, three independent functions p_1; p_2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative u_{xx}. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity (called g-monotonicity) properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG framework. The g-monotonicity gives the DG methods the ability to select the mathematically "correct" solution (i.e., the viscosity solution) among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1401-m4357}, url = {http://global-sci.org/intro/article_detail/jcm/9873.html} }
TY - JOUR T1 - Mixed Interior Penalty Discontinuous Galerkin Methods for One-Dimensional Fully Nonlinear Second Order Elliptic and Parabolic Equations AU - Xiaobing Feng & Thomas Lewis JO - Journal of Computational Mathematics VL - 2 SP - 107 EP - 135 PY - 2014 DA - 2014/04 SN - 32 DO - http://doi.org/10.4208/jcm.1401-m4357 UR - https://global-sci.org/intro/article_detail/jcm/9873.html KW - Fully nonlinear PDEs KW - Viscosity solutions KW - Discontinuous Galerkin methods AB -

This paper is concerned with developing accurate and ecient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin (IP-DG) methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative u_{xx} of the solution u, three independent functions p_1; p_2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative u_{xx}. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity (called g-monotonicity) properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG framework. The g-monotonicity gives the DG methods the ability to select the mathematically "correct" solution (i.e., the viscosity solution) among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.

Xiaobing Feng & Thomas Lewis. (1970). Mixed Interior Penalty Discontinuous Galerkin Methods for One-Dimensional Fully Nonlinear Second Order Elliptic and Parabolic Equations. Journal of Computational Mathematics. 32 (2). 107-135. doi:10.4208/jcm.1401-m4357
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