The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H^2 norm and a L^2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation.

翻译：本文针对双调和方程提出并分析了广义弱Galerkin（gWG）有限元方法。在gWG格式中引入了一种新的广义离散弱二阶偏导数，允许在一般多边形或多面体单元内部及边界上定义的分片多项式函数进行任意组合。基于离散H^2范数和L^2范数建立了数值逼近的误差估计。数值结果验证了所提出的gWG方法在求解双调和方程时的精度和灵活性。