TY - JOUR
T1 - Orthogonal Spline Collocation for Singularly Perturbed Reaction Diffusion Problems in One Dimension
AU - Mishra , Pankaj
AU - Sharma , Kapil K.
AU - Pani , Amiya K.
AU - Fairweather , Graeme
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 647
EP - 667
PY - 2019
DA - 2019/02
SN - 16
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/13019.html
KW - Singularly perturbed reaction diffusion problems, orthogonal spline collocation, Shishkin mesh, quasi-optimal global error estimates, superconvergence.
AB - <p style="text-align: justify;">An orthogonal spline collocation method (OSCM) with $C^1$<span style="font-size: 13.3333px;"> </span>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 $H$<sup>$m$</sup> norms for $m$ = 1, 2 and in a discrete $L$<sup>2</sup>-norm are derived. These estimates are valid
uniformly with respect to the perturbation parameter. The results of numerical experiments are
presented for $C$<sup>1</sup> cubic splines ($r$ = 3) and $C$<sup>1</sup> quintic splines ($r$ = 5) to demonstrate the efficacy
of the OSCM and confirm our theoretical findings. Further, quasi-optimal a $priori$ estimates in $L$<sup>2</sup>, $L$<sup>∞</sup> and $W$<sup>1</sup><sup>,∞</sup>-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.</p>