Year: 2019
Author: Pankaj Mishra, Kapil K. Sharma, Amiya K. Pani, Graeme Fairweather
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 4 : pp. 647–667
Abstract
An orthogonal spline collocation method (OSCM) with $C^1$ splines of degree $r$ ≥ 3 is analyzed for the numerical solution of singularly perturbed reaction diffusion problems in one dimension. The method is applied on a Shishkin mesh and quasi-optimal error estimates in weighted $H$$m$ norms for $m$ = 1, 2 and in a discrete $L$2-norm are derived. These estimates are valid uniformly with respect to the perturbation parameter. The results of numerical experiments are presented for $C$1 cubic splines ($r$ = 3) and $C$1 quintic splines ($r$ = 5) to demonstrate the efficacy of the OSCM and confirm our theoretical findings. Further, quasi-optimal a $priori$ estimates in $L$2, $L$∞ and $W$1,∞-norms are observed in numerical computations. Finally, superconvergence of order 2$r$ − 2 at the mesh points is observed in the approximate solution and also in its first derivative when $r$ = 5.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2019-IJNAM-13019
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 4 : pp. 647–667
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Singularly perturbed reaction diffusion problems orthogonal spline collocation Shishkin mesh quasi-optimal global error estimates superconvergence.