The symmetric $C^0$ interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh-refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. Two appendices present the realization of our proposed parameter selection in various established finite element software packages as well as a detailed documentation of a self-contained MATLAB program for the lowest-order $C^0$ interior penalty method.

翻译：对称$C^0$内罚方法是求解双调和方程最常用的间断伽辽金方法之一。本文针对任意多项式次数，提出了一种基于底层三角剖分几何结构的稳定性参数自动局部选取策略。所提选取方式能确保离散格式的稳定性，并保证离散椭圆性常数具有可靠下界。针对均匀自适应网格加密及多种多项式次数的数值实验验证了该局部参数选取的可靠性与高效性，并建议在实际应用中采用。该方法在二维三角形网格上进行了文档化实现，但其核心思想可推广至高维空间、非均匀多项式次数及矩形离散格式。附录分别展示了如何在多种成熟有限元软件包中实现所提参数选取方案，以及一套用于最低阶$C^0$内罚方法的自包含MATLAB程序的完整文档说明。