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Finite elements using a plane-wave basis for scattering of surface water waves

A generalized, frequency–independent, finite–element method is proposed for the numerical simulation of combined diffraction, refraction, radiation, reflection and absorption of water waves. Plane waves are used as an external function space, exploiting the standard set of approximation functions defined by the finite element method to enforce conformity and to impose boundary conditions in a straightforward manner by applying the method to the plane waves on each element. An integration method originally devised only for diffraction of waves by Ortiz and Sanchez in 2001 is extended to combined refraction and diffraction of high–frequency waves, retaining its dramatic savings in the number of operations for a given error compared with standard numerical integration methods.