Title: Very high-order symmetric positive-interior quadrature rules on triangles and tetrahedra Show Chinese title

Title: 非常高阶对称正内部求积公式在三角形和四面体上

Abstract: We present novel fully-symmetric quadrature rules with positive weights and strictly interior nodes of degrees up to 84 on triangles and 40 on tetrahedra. Initial guesses for solving the nonlinear systems of equations needed to derive quadrature rules are generated by forming tensor-product structures on quadrilateral/hexahedral subdomains of the simplices using the Legendre-Gauss nodes on the first half of the line reference element. In combination with a methodology for node elimination, these initial guesses lead to the development of highly efficient quadrature rules, even for very high polynomial degrees. Using existing estimates of the minimum number of quadrature points for a given degree, we show that the derived quadrature rules on triangles and tetrahedra are more than 95% and 80% efficient, respectively, for almost all degrees. The accuracy of the quadrature rules is demonstrated through numerical examples. Show Chinese abstract

Abstract: 我们提出了新颖的完全对称求积规则，其权重为正，节点严格位于三角形上的度数高达84，四面体上的度数高达40。 为了解决推导求积规则所需的非线性方程组，通过在单形的四边形/六面体子域上形成张量积结构，并使用线参考元前半部分的勒让德-高斯节点来生成初始猜测。 结合节点消除的方法，这些初始猜测导致了高度高效的求积规则，即使对于非常高的多项式次数也是如此。 利用现有给定度数的最小求积点数的估计，我们表明，对于几乎所有度数，三角形和四面体上的推导求积规则的效率分别超过95%和80%。 通过数值例子展示了求积规则的准确性。