Enrichment strategies for the simplicial linear finite elements

In this paper, we introduce a new class of finite elements by enriching the standard simplicial linear finite element in Rd with additional functions which are not necessarily polynomials. We provide necessary and sufficient conditions on the enrichment functions, which guarantee the existence of families of such enriched elements. Furthermore, we derive explicit formulas for their associated basis functions. We also show that the approximation error, obtained by using the proposed enriched elements, can be written as the error of the standard simplicial linear finite element plus a second term which depends on the enrichment functions. By using this decomposition, we derive explicit bounds in both L∞-norm and L1-norm.