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 - <p style="text-align: justify;">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.</p>