Locally enriched finite elements for the Helmholtz equation in two dimensions

This paper presents a finite element method for the solution of Helmholtz problems at high wave numbers that offers the potential of capturing many wavelengths per nodal spacing. This is done by constructing oscillatory shape functions as the product of polynomial shape functions and either Bessel functions or planar waves. The resulting elementary matrices obtained from the Galerkin–Bubnov formulation contain oscillatory terms and are evaluated using high order Gauss–Legendre integration. The problem of interest deals with the diffraction of an incident plane wave by a rigid circular cylinder. Numerical experiments are carried out on a square computational domain for which the analytical solution of the problem is imposed on its boundary. The obtained results using the proposed finite element models are compared for different locations of the computational domain, with respect to the diffracting object, and for increasing wave number. It is shown that in the near field, the plane wave basis finite element model provides more accurate results. However, far from the scattering object, the Bessel function approximating model provides better accuracy.