Finite element analysis of two-dimensional non-linear transient water waves

The two-dimensional nonlinear time domain free surface flow problem is analyzed by the finite element method. Two approaches are used. One is based on the velocity potential which is approximated by means of shape functions. The solution is obtained through use of a variational statement, and the velocity is obtained subsequently by the Galerkin method. The other approach is to write both potential and velocity in terms of the shape functions at the same time. Their solutions are derived from the same equation by using another variational statement. Numerical results are given for the vertical wave maker problem and for a transient wave in a rectangular container. They are compared with analytical solutions, and very good agreement is found.