The boundary layer problem: A fourth-order adaptive collocation approach

A finite element approach, based on the cubic BB-spline collocation, is presented for the numerical solution of a class of singularly perturbed two-point boundary value problems that possesses a boundary layer at one or two end points. Due to the existence of a layer, the problem is handled using an adaptive spline collocation approach constructed over a non-uniform Shishkin-like mesh, defined via a carefully selected generating function. To tackle the case of nonlinearity, if it exists, an iterative scheme arising from Newton’s method is employed.The rate of convergence is verified to be of fourth-order and is calculated using the double-mesh principle. The efficiency and applicability of the method are demonstrated by applying it to a number of linear and nonlinear examples. The numerical solutions are compared with both analytical and other existing numerical solutions in the literature. The numerical results confirm that this method is superior when contrasted with other accessible approaches and yields more accurate solutions.