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Volume 31, Issue 6
A New Direct Discontinuous Galerkin Method with Symmetric Structure for Nonlinear Diffusion Equations

Chad Vidden & Jue Yan

DOI: 10.4208/jcm.1307-m4273

J. Comp. Math., 31 (2013), pp. 638-662.

Published online: 2013-12

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  • Abstract

In this paper we continue the study of discontinuous Galerkin finite element methods for nonlinear diffusion equations following the direct discontinuous Galerkin (DDG) methods for diffusion problems [17] and the direct discontinuous Galerkin (DDG) methods for diffusion with interface corrections [18]. We introduce a numerical flux for the test function, and obtain a new direct discontinuous Galerkin method with symmetric structure. Second order derivative jump terms are included in the numerical flux formula and explicit guidelines for choosing the numerical flux are given. The constructed scheme has a symmetric property and an optimal $L^2(L^2)$ error estimate is obtained. Numerical examples are carried out to demonstrate the optimal $(k+1)$th order of accuracy for the method with $P^k$ polynomial approximations for both linear and nonlinear problems, under one-dimensional and two-dimensional settings.

  • Keywords

Discontinuous Galerkin Finite Element method, Diffusion equation, Stability, Convergence.

  • AMS Subject Headings

65M60.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-31-638, author = {Chad Vidden and Jue Yan}, title = {A New Direct Discontinuous Galerkin Method with Symmetric Structure for Nonlinear Diffusion Equations}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {6}, pages = {638--662}, abstract = {

In this paper we continue the study of discontinuous Galerkin finite element methods for nonlinear diffusion equations following the direct discontinuous Galerkin (DDG) methods for diffusion problems [17] and the direct discontinuous Galerkin (DDG) methods for diffusion with interface corrections [18]. We introduce a numerical flux for the test function, and obtain a new direct discontinuous Galerkin method with symmetric structure. Second order derivative jump terms are included in the numerical flux formula and explicit guidelines for choosing the numerical flux are given. The constructed scheme has a symmetric property and an optimal $L^2(L^2)$ error estimate is obtained. Numerical examples are carried out to demonstrate the optimal $(k+1)$th order of accuracy for the method with $P^k$ polynomial approximations for both linear and nonlinear problems, under one-dimensional and two-dimensional settings.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1307-m4273}, url = {http://global-sci.org/intro/article_detail/jcm/9758.html} }
TY - JOUR T1 - A New Direct Discontinuous Galerkin Method with Symmetric Structure for Nonlinear Diffusion Equations AU - Chad Vidden & Jue Yan JO - Journal of Computational Mathematics VL - 6 SP - 638 EP - 662 PY - 2013 DA - 2013/12 SN - 31 DO - http://doi.org/10.4208/jcm.1307-m4273 UR - https://global-sci.org/intro/article_detail/jcm/9758.html KW - Discontinuous Galerkin Finite Element method, Diffusion equation, Stability, Convergence. AB -

In this paper we continue the study of discontinuous Galerkin finite element methods for nonlinear diffusion equations following the direct discontinuous Galerkin (DDG) methods for diffusion problems [17] and the direct discontinuous Galerkin (DDG) methods for diffusion with interface corrections [18]. We introduce a numerical flux for the test function, and obtain a new direct discontinuous Galerkin method with symmetric structure. Second order derivative jump terms are included in the numerical flux formula and explicit guidelines for choosing the numerical flux are given. The constructed scheme has a symmetric property and an optimal $L^2(L^2)$ error estimate is obtained. Numerical examples are carried out to demonstrate the optimal $(k+1)$th order of accuracy for the method with $P^k$ polynomial approximations for both linear and nonlinear problems, under one-dimensional and two-dimensional settings.

Chad Vidden and Jue Yan. (2013). A New Direct Discontinuous Galerkin Method with Symmetric Structure for Nonlinear Diffusion Equations. Journal of Computational Mathematics. 31 (6). 638-662. doi:10.4208/jcm.1307-m4273
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