  Weak Galerkin methods refer to general finite element methods for PDEs in
which differential operators are approximated by their weak forms as
distributions. Such weak forms give rise to desirable flexibilities in
enforcing boundary and interface conditions. A weak Galerkin finite element
method (WG-FEM) is developed in this paper for solving elliptic partial
differential equations (PDEs) with discontinuous coefficients and interfaces.
The paper also presents many numerical tests for validating the WG-FEM for
solving second order elliptic interface problems. For such interface problems,
the solution possesses a certain singularity due to the nonsmoothness of the
interface. A challenge in research is to design high order numerical methods
that work well for problems with low regularity in the solution. The best known
numerical scheme in the literature is of order one for the solution itself in
$L_\infty$ norm. It is demonstrated that the WG-FEM of lowest order is capable
of delivering numerical approximations that are of order 1.75 in the usual
$L_\infty$ norm for $C^1$ or Lipschitz continuous interfaces associated with a
$C^1$ or $H^2$ continuous solutions. Theoretically, it is proved that high
order of numerical schemes can be designed by using the WG-FEM with polynomials
of high order on each element.
