Several types of simultaneous approximation term (SAT) for diffusion problems discretized with multidimensional summation-by-parts (SBP) operators are analyzed based on a common framework. Conditions under which the SBP-SAT discretizations are consistent, conservative, adjoint consistent, and energy stable are presented. For SATs leading to primal and adjoint consistent discretizations, the error in output functionals is shown to be of order $h^{2p}$ when a degree $p$ multidimensional SBP operator is used to discretize the spatial derivatives. SAT penalty coefficients corresponding to various discontinuous Galerkin fluxes developed for elliptic partial differential equations are identified. We demonstrate that the original method of Bassi and Rebay, the modified method of Bassi and Rebay, and the symmetric interior penalty method are equivalent when implemented with SBP diagonal-E operators that have diagonal norm matrix, e.g., the Legendre-Gauss-Lobatto SBP operator in one space dimension. Similarly, the local discontinuous Galerkin and the compact discontinuous Galerkin schemes are equivalent for this family of operators. The analysis remains valid on curvilinear grids if a degree $\le p+1$ bijective polynomial mapping from the reference to physical elements is used. Numerical experiments with the two-dimensional Poisson problem support the theoretical results.
翻译:在共同框架的基础上,分析了SBP-SAT离散、保守、联合一致和能源稳定的条件。对于导致原始和联合一致离散的SAT,输出功能中的误差为$h ⁇ 2p},当使用某种程度的多维SBP操作员将空间衍生物分解时,产出功能中的误差为$$p$的多维SBP操作员用一个空间层面的分解。同样,与为椭圆部分差异方程式开发的各种不连续性加热金通量相对应的SAT惩罚系数。我们证明,巴西和雷拜的原始方法、加西和雷拜的修改方法和对等性内部处罚方法在与SBPBP有对等规范矩阵的Sgoag-E操作员一起实施时,即为 $oqual-Gaus-Lubas-Lubadto SBBP操作员在一个空间层面的分数。同样,当地不连续加热金和紧固基加热金计算器的参照率计划,如果用于这一家庭等级的计算器的双轨模型,那么,则对数字的内测算值的计算器的双基的计算器的计算器的分析为等。