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Volume 33, Issue 3
Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations

Yang Yang & Chi-Wang Shu

DOI: 10.4208/jcm.1502-m2014-0001

J. Comp. Math., 33 (2015), pp. 323-340.

Published online: 2015-06

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  • Abstract

In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the LDG solution and the exact solution is ($k$+2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is ($k$+2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for $P^k$ polynomials with arbitrary $k$ ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.

  • Keywords

Superconvergence, Local discontinuous Galerkin method, Parabolic equation, Initial discretization, Error estimates, Radau points.

  • AMS Subject Headings

65M60, 65M15.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yyang7@mtu.edu (Yang Yang)

shu@dam.brown.edu (Chi-Wang Shu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-33-323, author = {Yang , Yang and Shu , Chi-Wang}, title = {Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {3}, pages = {323--340}, abstract = {

In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the LDG solution and the exact solution is ($k$+2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is ($k$+2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for $P^k$ polynomials with arbitrary $k$ ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1502-m2014-0001}, url = {http://global-sci.org/intro/article_detail/jcm/9845.html} }
TY - JOUR T1 - Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations AU - Yang , Yang AU - Shu , Chi-Wang JO - Journal of Computational Mathematics VL - 3 SP - 323 EP - 340 PY - 2015 DA - 2015/06 SN - 33 DO - http://doi.org/10.4208/jcm.1502-m2014-0001 UR - https://global-sci.org/intro/article_detail/jcm/9845.html KW - Superconvergence, Local discontinuous Galerkin method, Parabolic equation, Initial discretization, Error estimates, Radau points. AB -

In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the LDG solution and the exact solution is ($k$+2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is ($k$+2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for $P^k$ polynomials with arbitrary $k$ ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.

Yang , Yang and Shu , Chi-Wang. (2015). Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations. Journal of Computational Mathematics. 33 (3). 323-340. doi:10.4208/jcm.1502-m2014-0001
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