论文标题
同时近似项和功能准确性，用于通过逐行的多维总和离散的扩散问题
Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summation-by-parts operators
论文作者
论文摘要
基于公共框架分析了几种类型的同时近似项（SAT），该近似问题是通过对角线数多维总和（SBP）运算符离散的。 SBP-SAT离散化是一致，保守，伴随一致和能量稳定的条件。对于导致原始和伴随一致离散的SAT，当使用度$ p $多维SBP操作员用于离散空间衍生词时，输出功能的错误显示为$ h^{2p} $。确定了与针对椭圆形偏微分方程开发的各种不连续的Galerkin通量相对应的SAT惩罚系数。我们证明，Bassi和Rebay的原始方法，Bassi和Rebay的修改方法以及对称内部惩罚方法是等效的，当使用具有对角线Norm矩阵的SBP对角线运算符，例如Legendre-Gauss-Gauss-Gauss-gauss-lobatto sbp sbp Operenter。同样，当地的不连续的盖尔金和紧凑的不连续的盖金条方案对于这个运营商家族而言是等效的。如果使用$ \ le p+1 $ bi原始元素映射，则使用从参考物理元素的参考元素进行$ \ le p+1 $ p+1 $的曲线网格。二维泊松问题的数值实验支持理论结果。
Several types of simultaneous approximation term (SAT) for diffusion problems discretized with diagonal-norm 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.
