 
Summary: Variational Spacetime (Dis)continuous Galerkin
Method for Linear Free Surface Waves
V.R. Ambati, J.J.W. van der Vegt, and O. Bokhove
v.r.ambati@math.utwente.nl, o.bokhove@math.utwente.nl and
j.j.w.vandervegt@math.utwente.nl
Numerical Analysis and Computational Mechanics Group
Department of Applied Mathematics, University of Twente
P.O. Box 217, Enschede, The Netherlands.
March 5, 2008
Abstract
A new variational (dis)continuous Galerkin finite element method is presented for linear free
surface gravity water wave equations. In this method, the spacetime finite element discretiza
tion is based on a discrete variational formulation analogous to a version of Luke's variational
principle. The finite element discretization results into a linear algebraic system of equations
with a symmetric and compact stencil. These equations have been solved using the PETSc
package, in which a block sparse matrix storage routine is used to build the matrix and an
efficient conjugate gradient solver to solve the equations. The finite element scheme is verified
against exact solutions: linear free surface waves in a periodic domain and ones generated by a
harmonic wave maker in a rectangular wave basin. We found that the variational scheme has
no dissipation and minimal dispersion errors in the wave propagation, and that the numerical
