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On the Numerical Solution for Singularly Perturbed Second-order ODEs

In this article we consider the approximation of singularly perturbed boundary value problems using a local adaptive grid h-refinement for finite element method, the variation iteration method and the homotopy perturbation method. The solution to such problems contains boundary layers which overlap and interact and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. The results obtained by these methods are compared to the exact solutions for some model problems. It is found that the local adaptive grid h-refinement for finite element method is highly stable methods and always converges to the solution independently on the singular perturbation parameters.