Computation of three-dimensional standing water waves

We develop a method for computing three-dimensional gravity-driven water waves, which we use to search for time-periodic standing wave solutions. We simulate an inviscid, irrotational, incompressible fluid bounded below by a flat wall, and above by an evolving free surface. The computations make use of spectral derivatives on the surface, but also require computing a velocity potential in the bulk, which we carry out using a finite element method with fourth-order elements that are curved to match the free surface. This computationally expensive step is solved using a parallel multigrid algorithm, which is discussed in detail. Time-periodic solutions are searched for using a previously developed overdetermined shooting method. Several families of large-amplitude three-dimensional standing waves are found in both shallow and deep regimes, and their physical characteristics are examined and compared to previously known two-dimensional solutions.