Hierarchical enrichment for bridging scales and mesh-free boundary conditions

The finite-element method, when used with a basis made up of piecewise polynomials, often requires the generation of a very fine computational mesh in order to capture localized solution phenomena such as boundary layers or near-singularities. Enrichment of the basis with additional functions, obtained through analytical or experimental means, can allow for a coarser mesh and more accurate solution. We introduce an enrichment scheme in which an interaction or `bridging' scale term is used to separate the basis formed by the enrichment functions from the original set of basis functions, in effect making the enrichment hierarchical. This separation of scales allows the simple application of essential boundary conditions. It also allows a quantification of the effects of the enrichment, leading to strategies for error estimation and control of the stiffness matrix condition number. We also find that this formulation allows for the simple application of essential boundary conditions for mesh-free shape functions, which are notoriously problematic. We find that for multiple dimensions, care must be taken to derive a weak form which is truly consistent with the strong form on the essential boundary.