TY - JOUR T1 - Finite Volume Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems 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.