An improved finite element space for discontinuous pressures

We consider incompressible Stokes flow with an internal interface at which the pressure is discontinuous, as happens for example in problems involving surface tension. We assume that the mesh does not follow the interface, which makes classical interpolation spaces to yield suboptimal convergence rates (typically, the interpolation error in the L 2 ( Ω ) -norm is of order h 1 2 ). We propose a modification of the P 1 -conforming space that accommodates discontinuities at the interface without introducing additional degrees of freedom or modifying the sparsity pattern of the linear system. The unknowns are the pressure values at the vertices of the mesh and the basis functions are computed locally at each element, so that the implementation of the proposed space into existing codes is straightforward. With this modification, numerical tests show that the interpolation order improves to O h 3 2 . w pressure space is implemented for the stable P 1 + / P 1 mini-element discretization, and for the stabilized equal-order P 1 / P 1 discretization. Assessment is carried out for Poiseuille flow with a forcing surface and for a static bubble. In all cases the proposed pressure space leads to improved convergence orders and to more accurate results than the standard P 1 space. In addition, two Navier–Stokes simulations with moving interfaces (Rayleigh–Taylor instability and merging bubbles) are reported to show that the proposed space is robust enough to carry out realistic simulations.