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美元 内置惩罚方法是双声调方程中最受欢迎的不连续加列金方法之一。本文介绍了从任意多面度的底三角对地测量中自动选择相关稳定性参数。 拟议的选择确保了稳定的离散性,并有保证的离散椭圆常数。 统一和适应性网状精炼和各种多面度的数值证据支持了本地参数选择的可靠性和效率,并在实际中推荐了这一点。 该方法以2D形式记录在三角形中,但后面的方法可以推广到更高的维度、非统一多面度和矩形离异性。 有两个附录介绍了我们在各种固定的有限元素软件包中实现拟议参数选择的情况,以及一个自成一体的MATLAB程序的详细文件,用于最低级的 $C$0美元内部罚款方法。</s>