A Statically Condensable Enrichment for Pressure Discontinuities in Two-Phase Flows

Abstract

We introduce a new finite element space for discontinuous pressures at immersed boundaries not conforming with the mesh. The proposed space incorporates two additional degrees of freedom that are local to the elements crossed by the interface, linear on each side, discontinuous at the interface and zero at the element nodes. The new degrees of freedom can be statically condensed before final assembly, therefore avoiding difficulties associated with the update of the mesh graph in the case of moving interfaces as happens for instance with the well known extended finite element method (XFEM).
The implementation of the new space in any existing finite element code is extremely easy in two and three spatial dimensions, since the new shape functions are based on the usual P1 functions. The new space is compared with the classical P1-conforming space and with another finite element space without additional unknowns also proposed by the authors (see Ausas, Simeoni and Buscaglia, Comput. Meth. Appl. Mech. Engng., 2010) in several problems involving jumps in the viscosity and in the presence of singular forces, in two dimensions and in more challenging three dimensional situations. Based on
the numerical experiments we show that the behavior of the new space is equal or better than that of the aforementioned space.