The weak element method applied to Helmholtz type equations

Abstract

Helmholtz type boundary value problems are important in a variety of scattering and diffraction problems. Standard numerical schemes based on finite difference, finite element, or integral equation methods are generally not well suited for these problems in the ‘intermediate frequency range’ since the oscillatory solution is not accurately approximated by piecewise polynomials. In this paper, a version of the weak element method is employed to numerically solve these problems in two dimensions. This method consist of partitioning the domain into small ‘elements’ and locally approximating the solution in each element by a sum of exponentials. These piecewise approximations are joined together at interelement boundaries by continuity conditions for certain functionals of the approximate solution. The method is analyzed using a complementary variational formulation. It is shown that the weak element method is considerably more accurate than standard discretization methods when the solution is adequately approximated locally by the exponential basis functions. These results are validated by numerical experiments.