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Volume 31, Issue 5
Finite Volume Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems

Waixiang Cao, Zhimin Zhang & Qingsong Zou

DOI: 10.4208/jcm.1304-m4280

J. Comp. Math., 31 (2013), pp. 488-508.

Published online: 2013-10

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  • Abstract

We analyze finite volume schemes of arbitrary order $r$ for the one-dimensional singularly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as $(N^{-1}ln(N+1))^r$, where $2N$ is the number of subintervals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order $(N^{-1}ln(N+1))^{2r}$, while at the Gauss points, the derivative error super-converges with order $(N^{-1}ln(N+1))^{r+1}$. All the above convergence and superconvergence properties are independent of the perturbation parameter $ε$. Numerical results are presented to support our theoretical findings.

  • Keywords

Finite Volume, High Order, Superconvergence, Convection-Diffusion.

  • AMS Subject Headings

65N30, 65N12, 65N06.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-31-488, author = {Waixiang Cao, Zhimin Zhang and Qingsong Zou}, title = {Finite Volume Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {5}, pages = {488--508}, abstract = {

We analyze finite volume schemes of arbitrary order $r$ for the one-dimensional singularly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as $(N^{-1}ln(N+1))^r$, where $2N$ is the number of subintervals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order $(N^{-1}ln(N+1))^{2r}$, while at the Gauss points, the derivative error super-converges with order $(N^{-1}ln(N+1))^{r+1}$. All the above convergence and superconvergence properties are independent of the perturbation parameter $ε$. Numerical results are presented to support our theoretical findings.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1304-m4280}, url = {http://global-sci.org/intro/article_detail/jcm/9749.html} }
TY - JOUR T1 - Finite Volume Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems AU - Waixiang Cao, Zhimin Zhang & Qingsong Zou JO - Journal of Computational Mathematics VL - 5 SP - 488 EP - 508 PY - 2013 DA - 2013/10 SN - 31 DO - http://doi.org/10.4208/jcm.1304-m4280 UR - https://global-sci.org/intro/article_detail/jcm/9749.html KW - Finite Volume, High Order, Superconvergence, Convection-Diffusion. AB -

We analyze finite volume schemes of arbitrary order $r$ for the one-dimensional singularly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as $(N^{-1}ln(N+1))^r$, where $2N$ is the number of subintervals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order $(N^{-1}ln(N+1))^{2r}$, while at the Gauss points, the derivative error super-converges with order $(N^{-1}ln(N+1))^{r+1}$. All the above convergence and superconvergence properties are independent of the perturbation parameter $ε$. Numerical results are presented to support our theoretical findings.

Waixiang Cao, Zhimin Zhang and Qingsong Zou. (2013). Finite Volume Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems. Journal of Computational Mathematics. 31 (5). 488-508. doi:10.4208/jcm.1304-m4280
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