An Ε-uniform Finite Element Method for Singularly Perturbed Boundary Value Problems

This work develops an ε-uniform finite element method for singularly perturbed boundary value problems. A surprising and remarkable observation is illustrated: By inserting one node arbitrarily in any element, the new finite element solution always intersects with the original one at fixed points, and the errors at those points converge at the same rate as regular boundary value problems (without boundary layers). Using this fact, an effective ε-uniform approximation out of boundary layer is proposed by adding one point only in the element that contains the boundary layer. The thickness of the boundary layer need not be known a priori. Numerical results are carried out and compared to the Shishkin mesh for demonstration purpose.