The Generalized S-Mesh Based Hybrid Algorithm for Singularly Perturbed Problems With Interior Layer

Abstract

In this article we present the analysis of generalized S-mesh (denoted by $$S(\ell )$$ ) based hybrid algorithm for a class of second-order singularly perturbed differential equations with discontinuous convection coefficient. The solution of the problem exhibits interior layer because of the discontinuity in convection coefficient. We have demonstrated the generation of $$S(\ell )$$ mesh for a domain with interior layer and also estimated that the algorithm is perturbation parameter ( $$\epsilon $$ ) uniformly convergent with error asymptotic to $$N^{-2}(\ln ^\ell (N))^2$$ where $$ \ell \ll N ~ \& ~ \ell \in {\mathbb {N}}$$ . The numerical experiments of the algorithm on Shishkin mesh, B-type mesh and $$S(\ell )$$ mesh are carried out for couple of examples, to demonstrate the efficiency of the proposed hybrid algorithm on $$S(\ell )$$ mesh. It was observed that upwind algorithm on B-type mesh is more efficient than upwind algorithm on Shishkin or $$S(\ell )$$ mesh and the proposed hybrid algorithm on $$S(\ell )$$ mesh is of second order and observed to be efficient.