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Volume 10, Issue 2
Local Error Estimates of the LDG Method for 1-D Singularly Perturbed Problems

H. Zhu & Z. Zhang

Int. J. Numer. Anal. Mod., 10 (2013), pp. 350-373.

Published online: 2013-10

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  • Abstract

In this paper local discontinuous Galerkin method (LDG) was analyzed for solving 1-D convection-diffusion equations with a boundary layer near the outflow boundary. Local error estimates are established on quasi-uniform meshes with maximum mesh size $h$. On a subdomain with $O(h\ln(1/h))$ distance away from the outflow boundary, the $L^2$ error of the approximations to the solution and its derivative converges at the optimal rate $O(h^{k+1})$ when polynomials of degree at most $k$ are used. Numerical experiments illustrate that the rate of convergence is uniformly valid and sharp. The numerical comparison of the LDG method and the streamline-diffusion finite element method are also presented.

  • Keywords

Local discontinuous Galerkin method, singularly perturbed, local error estimates.

  • AMS Subject Headings

65L10, 65L20, 65L60,65M50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-350, author = {}, title = {Local Error Estimates of the LDG Method for 1-D Singularly Perturbed Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {2}, pages = {350--373}, abstract = {

In this paper local discontinuous Galerkin method (LDG) was analyzed for solving 1-D convection-diffusion equations with a boundary layer near the outflow boundary. Local error estimates are established on quasi-uniform meshes with maximum mesh size $h$. On a subdomain with $O(h\ln(1/h))$ distance away from the outflow boundary, the $L^2$ error of the approximations to the solution and its derivative converges at the optimal rate $O(h^{k+1})$ when polynomials of degree at most $k$ are used. Numerical experiments illustrate that the rate of convergence is uniformly valid and sharp. The numerical comparison of the LDG method and the streamline-diffusion finite element method are also presented.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/572.html} }
TY - JOUR T1 - Local Error Estimates of the LDG Method for 1-D Singularly Perturbed Problems JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 350 EP - 373 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/572.html KW - Local discontinuous Galerkin method, singularly perturbed, local error estimates. AB -

In this paper local discontinuous Galerkin method (LDG) was analyzed for solving 1-D convection-diffusion equations with a boundary layer near the outflow boundary. Local error estimates are established on quasi-uniform meshes with maximum mesh size $h$. On a subdomain with $O(h\ln(1/h))$ distance away from the outflow boundary, the $L^2$ error of the approximations to the solution and its derivative converges at the optimal rate $O(h^{k+1})$ when polynomials of degree at most $k$ are used. Numerical experiments illustrate that the rate of convergence is uniformly valid and sharp. The numerical comparison of the LDG method and the streamline-diffusion finite element method are also presented.

H. Zhu & Z. Zhang. (1970). Local Error Estimates of the LDG Method for 1-D Singularly Perturbed Problems. International Journal of Numerical Analysis and Modeling. 10 (2). 350-373. doi:
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