The impact of smooth W-grids in the numerical solution of singular perturbation two-point boundary value problems

This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter 1. For problems on [0,1] with a boundary layer at one end point, the local mesh width hi = xi+1 xi, with 0 = x0 < x1 < < xN = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise uniform Shishkin mesh. For small , neither the finite difference method nor the finite element method produces satisfactory results on the Shishkin mesh. By contrast, accuracy is vastly improved on theW-grid, which typically produces the nominal 2nd order behavior in L, for large as well as small values of N, and over a wide range of values of . We conclude that the smoothness of the mesh is of crucial importance to accuracy, efficiency and robustness. Published by Elsevier Inc.