This paper introduces an extension of the well-known Morley element for the biharmonic equation, extending its application from triangular elements to general polytopal elements using the weak Galerkin finite element methods. By leveraging the Schur complement of the weak Galerkin method, this extension not only preserves the same degrees of freedom as the Morley element on triangular elements but also expands its applicability to general polytopal elements. The numerical scheme is devised by locally constructing weak tangential derivatives and weak second-order partial derivatives. Error estimates for the numerical approximation are established in both the energy norm and the $L^2$ norm. A series of numerical experiments are conducted to validate the theoretical developments.

翻译：本文针对双调和方程，将经典的Morley元从三角形单元推广至一般多边形单元，利用弱Galerkin有限元方法实现了这一推广。通过弱Galerkin方法的Schur补，该推广不仅保留了Morley元在三角形单元上的相同自由度，还将其适用范围扩展至一般多边形单元。数值方案通过局部构造弱切向导数和弱二阶偏导数进行设计。在能量范数和$L^2$范数下建立了数值逼近的误差估计。一系列数值实验验证了理论结果的有效性。