Improved methods for the enrichment and analysis of the simplicial vector-valued linear finite elements

The simplicial vector linear finite elements are commonly employed for the numerical solution of the stationary Stokes equations. Nevertheless, they exhibit significant limitations when applied to more complex scenarios. In response to these shortcomings, Bernardi and Raugel introduced an enriched finite element which is a generalization of the conventional simplicial vector linear finite element. It employs polynomials as enrichment functions and its application extends across a broad spectrum of practical engineering computational fields. However, for some types of problems, these enrichment functions are not very efficient. The main goal of this paper is to introduce a comprehensive method for enhancing the simplicial vector linear finite element with non-polynomial enrichment functions. The enriched finite element is defined concerning any simplex and can be considered an extension of the Bernardi and Raugel finite element. A crucial component in this context is the characterization result, formulated regarding the nonvanishing of a specific determinant. This result provides both necessary and sufficient conditions for the existence of families of enriched elements. In conclusion, we present numerical tests that show the efficacy of the suggested enrichment strategy.