  This paper introduces a new weak Galerkin (WG) finite element method for
second order elliptic equations on polytopal meshes. This method, called
WG-FEM, is designed by using a discrete weak gradient operator applied to
discontinuous piecewise polynomials on finite element partitions of arbitrary
polytopes with certain shape regularity. The paper explains how the numerical
schemes are designed and why they provide reliable numerical approximations for
the underlying partial differential equations. In particular, optimal order
error estimates are established for the corresponding WG-FEM approximations in
both a discrete $H^1$ norm and the standard $L^2$ norm. Numerical results are
presented to demonstrate the robustness, reliability, and accuracy of the
WG-FEM. All the results are derived for finite element partitions with
polytopes. Allowing the use of discontinuous approximating functions on
arbitrary polytopal elements is a highly demanded feature for numerical
algorithms in scientific computing.
