Orthogonal Spline Collocation for Singularly Perturbed Reaction Diffusion Problems in One Dimension

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.