  This article introduces and analyzes a weak Galerkin mixed finite element
method for solving the biharmonic equation. The weak Galerkin method, first
introduced by two of the authors (J. Wang and X. Ye) in an earlier publication
for second order elliptic problems, is based on the concept of discrete weak
gradients. The method allows the use of completely discrete finite element
functions on partitions of arbitrary polygon or polyhedron. In this article,
the idea of weak Galerkin method is applied to discretize the Ciarlet-Raviart
mixed formulation for the biharmonic equation. In particular, an a priori error
estimation is given for the corresponding finite element approximations. The
error analysis essentially follows the framework of Babuska, Osborn, and
Pitkaranta and uses specially designed mesh-dependent norms. The proof is
technically tedious due to the discontinuous nature of the weak Galerkin finite
element functions. Some computational results are presented to demonstrate the
efficiency of the method.
