An edge-based pressure stabilization technique for finite elements on arbitrarily anisotropic meshes

In this paper, we analyze a stabilized equal-order finite element approximation for the Stokes equations on anisotropic meshes. In particular, we allow arbitrary anisotropies in a subdomain, for example, along the boundary of the domain, with the only condition that a maximum angle is fulfilled in each element. This discretization is motivated by applications on moving domains as arising, for example, in fluid-structure interaction or multiphase-flow problems. To deal with the anisotropies, we define a modification of the original continuous interior penalty stabilization approach. We show analytically the discrete stability of the method and convergence of order in the energy norm and in the L²-norm of the velocities. We present numerical examples for a linear Stokes problem and for a nonlinear fluid-structure interaction problem, which substantiate the analytical results and show the capabilities of the approach.