This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete $H^2$ norm and the usual $L^2$ norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension.

翻译：本文介绍了 Morley 单元的一种推广形式，用于逼近双调和方程的解。传统上，Morley 单元仅限于三角形单元上的分段二次多项式，本文的推广利用弱 Galerkin 有限元方法，能够适应更高次的多项式以及一般多面体单元的灵活性。通过利用弱 Galerkin 方法中的 Schur 补，该推广在保证数值解充分精度和稳定性的同时，实现了最少的局部自由度。数值方案采用了局部构造的弱切向导数和弱二阶偏导数，从而精确逼近双调和方程。为了评估数值近似的精度，我们在离散 $H^2$ 范数和通常的 $L^2$ 范数下建立了最优阶误差估计。此外，本文还给出了数值结果，以验证所提出理论并展示该推广方法的有效性。