Resolving Boundary Layers with Harmonic Extension Finite Elements

In recent years, the standard numerical methods for partial differential equations have been extended with variants that address the issue of domain discretisation in complicated domains. Sometimes similar requirements are induced by local parameter-dependent features of the solutions, for instance, boundary or internal layers. The adaptive reference elements are one way with which harmonic extension elements, an extension of the p-version of the finite element method, can be implemented. In combination with simple replacement rule-based mesh generation, the performance of the method is shown to be equivalent to that of the standard p-version in problems where the boundary layers dominate the solution. The performance over a parameter range is demonstrated in an application of computational asymptotic analysis, where known estimates are recovered via computational means only.