On the stability and interpolating properties of the Hierarchical Interface-enriched Finite Element Method

The Hierarchical Interface-enriched Finite Element Method (HIFEM) is a technique for solving problems containing discontinuities in the field gradient using finite element meshes that do not conform (match) the domain morphology. The method is suitable for analyzing problems with complex geometries or when such geometry is not known a priori. Contrary to the eXtended/Generalized Finite Element Method (X/GFEM), enrichments are placed on nodes created along interfaces, and a recursive enrichment strategy is used to resolve multiple discontinuities crossing single elements. In this manuscript we rigorously study the approximating properties and stability of HIFEM. A study on the enrichments’ polynomial order shows that the formulation does not pass the patch test as long as enrichments do not replicate the approximating properties of partition of unity shape functions. Regarding stability, we show that condition numbers of system matrices grow at the same rate as those of standard FEM—and without requiring a preconditioner. This intrinsic stability is accomplished by means of a proper construction of enrichment functions that are properly scaled as interfaces approach mesh nodes. We further demonstrate that, even without scaling, using a simple preconditioner recovers stability. The method's stability is further demonstrated on the modeling of challenging thermal and mechanical problems with complex morphologies.