An unfitted high-order spectral element method for incompressible Navier-Stokes equations with a free surface: The pressure problem

We propose a novel unfitted high-order spectral element solver for nonlinear water wave simulations based on an incompressible Navier-Stokes formulation with a free surface. The solver achieves its unfitted high-order accuracy from a basis of polynomial functions that are combined with the recently developed shifted boundary method. This unfitted method allows for a natural curvature representation, circumvents complex meshing and re-meshing, and avoids the small-cut-cell problem. For water wave problems, there are multiple boundaries that are complex, curved, moving, and deforming in time. Hence, they are suitable for unfitted methods like the shifted boundary approach. For this work, we solely consider the dynamic pressure problem of the complete Navier-Stokes formulation.