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This paper presents a new and efficient numerical algorithm for thebiharmonic equation by using weak Galerkin (WG) finite element methods. The WGfinite element scheme is based on a variational form of the biharmonic equationthat is equivalent to the usual $H^2$-semi norm. Weak partial derivatives andtheir approximations, called discrete weak partial derivatives, are introducedfor a class of discontinuous functions defined on a finite element partition ofthe domain consisting of general polygons or polyhedra. The discrete weakpartial derivatives serve as building blocks for the WG finite element method.The resulting matrix from the WG method is symmetric, positive definite, andparameter free. An error estimate of optimal order is derived in an$H^2$-equivalent norm for the WG finite element solutions. Error estimates inthe usual $L^2$ norm are established, yielding optimal order of convergence forall the WG finite element algorithms except the one corresponding to the lowestorder (i.e., piecewise quadratic elements). Some numerical experiments arepresented to illustrate the efficiency and accuracy of the numerical scheme.

An Efficient Numerical Scheme for the Biharmonic Equation by Weak Galerkin Finite Element Methods on Polygonal or Polyhedral Meshes