Above: least squares finite element computation of an inertial membrane swimming on the surface of a Newtonian fluid (Re=100) by propagating waves along its body. Randomly distributed particles are shown advecting with the flow on a contour plot of the vorticity. Although this swimmer is not very efficient, it does consistently move to the right at a rate of approximately 10% of its length per cycle. Below: the nodes of the grid move vertically to remain fitted to the domain. Piecewise quadratic C0 elements are used for each of the six variables in the computation (2 velocity components, 2 shear components, vorticity, and pressure). Triangles on the upper wall have curved boundaries. A multigrid preconditioned conjugate gradient method is used to solve the equations at each timestep (or at each iteration of a steady-state calculation).