Numerical Stability Analysis of a Pressure Space with Embedded Discontinuities

Abstract

A finite element space with embedded discontinuities has been recently introduced by the authors (R. Ausas et al, Comp. Meth. Appl. Mech. Engrg, 199:1019-1031, 2009; see also Mecánica Computacional 28:1131-1148, 2009). This space has the same unknowns as the linear, continuous finite element space, but is locally modified to accommodate discontinuities at an arbitrary interface, not coincident with the element boundaries. The motivation comes from Eulerian treatment of surface tension problems, in which there is a pressure jump at the interface. It has been shown both numerically and theoretically (G. Buscaglia and A. Agouzal, CILAMCE, 2009) that this space has good interpolation properties. Numerical examples have also shown that, when used as pressure space in a finite element formulation of the Navier-Stokes equations, the proposed space leads to accurate results. This suggests that inf–sup stability conditions are satisfied, but they are very hard to prove because the space depends on the (arbitrary) location of the interface. In this paper we briefly review the eigenproblems associated with the discrete inf–sup condition (D. Malkus, Int. J. Engng. Sci., 19:1299-1310, 1981) and use them to numerically assess the stability and convergence of the proposed space, considering both mixed (mini–element) and stabilized (equal–order) formulations.