Stabilized mixed displacement-pressure finite element formulation for linear hydrodynamic problems with free surfaces

A mixed displacement-pressure formulation of the Stokes problem for incompressible fluids with free surfaces is developed for modeling the propagation of gravity waves in liquids and their interaction with structures using a Lagrangian approach. We assume that fluid displacements are small, making convective effects negligible and approximate the fluid velocities from the time derivative of the displacements. The resulting finite element equations are discretized with equal order for both displacement and pressure terms, together with employing stabilization techniques that circumvent the inf-sup requirements. The stability and accuracy of the methodology is finally demonstrated by solving some classical problems of hydrodynamics with free surfaces, comparing the results with known analytical solutions.