Asymptotic Analysis and a Uniformly Convergent Numerical Method for Singular Perturbation Problems

Approximation methods for boundary problems for a fourth-order singularly perturbed partial differential equation (PDE) are studied. Using a suitable variable change, we reduce the problem to a second-order PDE system with coupled boundary conditions. Taking into account asymptotic expansions of the solutions, we discrete the resulting problem by a tailored finite point method. It is proved that the scheme converges uniformly with respect to the small parameter involved. Numerical results are consistent with the theoretical findings.