  Weak Galerkin (WG) refers to general finite element methods for partial
differential equations in which differential operators are approximated by weak
forms through the usual integration by parts. In particular, WG methods allow
the use of discontinuous finite element functions in the algorithm design. One
of such examples was recently introduced by Wang and Ye for solving second
order elliptic problems. The goal of this paper is to apply the WG method of
Wang and Ye to the Helmholtz equation with high wave numbers. Several test
scenarios are designed for a numerical investigation on the accuracy,
convergence, and robustness of the WG method in both inhomogeneous and
homogeneous media over convex and non-convex domains. Our numerical experiments
indicate that weak Galerkin is a finite element technique that is easy to
implement, and provides very accurate and robust numerical solutions for the
Helmholtz problem with high wave numbers.
