Troesch’s problem: A B-spline collocation approach

A finite-element approach, based on cubic B-spline collocation, is presented for the numerical solution of Troesch’s problem. The method is used on both a uniform mesh and a piecewise-uniform Shishkin mesh, depending on the magnitude of the eigenvalues. This is due to the existence of a boundary layer at the right endpoint of the domain for relatively large eigenvalues. The problem is also solved using an adaptive spline collocation approach over a non-uniform mesh via exploiting an iterative scheme arising from Newton’s method.The convergence analysis is discussed and is shown to depend on the eigenvalues; in particular, the rate of convergence is calculated using the double-mesh principle. To demonstrate the efficiency of the method, a number of special cases are considered. The numerical solutions are compared with both the analytical solutions and other existing numerical solutions in the literature. It is observed that the results obtained by this method are quite satisfactory and accurate, and the method is applicable for a wide range of cases when contrasted with other available solutions.