Abstract Higher order finite element (FE) methods provide significant advantages in a number of applications such as wave propagation, where high order shape functions help to mitigate pollution (dispersion) error. However, classical assembly of higher order systems is computationally burdensome, requiring the evaluation of many point quadrature schemes. When the Discontinuous Petrov-Galerkin (DPG) FE methodology is employed, the use of an enriched test space further increases the computational burden of system assembly, increasing the relevance of improved assembly techniques. Sum factorization—a technique that exploits the tensor-product structure of shape functions to accelerate numerical integration—was proposed in Ref. [10] for the assembly of DPG matrices on hexahedral elements that reduced the computational complexity from order O ( p 9 ) to O ( p 7 ) (where p denotes polynomial order). In this work we extend the concept of sum factorization to the construction of DPG matrices on prismatic elements by expressing prism shape functions as tensor products of 2D simplex and 1D interval shape functions. Unexpectedly, the resulting sum factorization routines on partially-tensorized prism shape functions achieve the same O ( p 7 ) complexity as sum factorization on fully-tensorized hexahedra shape functions (as products of 1D interval shape functions) presented in Ref. [10]. Throughout this work we adhere to the theory of exact sequence energy spaces, proposing sum factorization routines for each of the 3D FE exact sequence energy spaces—H1, H(curl), H(div), and L2. Computational results for construction of the DPG Gram matrix on a prismatic element in each exact sequence energy space are presented, corroborating the expected O ( p 7 ) complexity. Additionally, construction of the DPG system for an ultraweak Maxwell problem on a prismatic element is considered and a partially-tensorized sum factorization for hexahedral elements is proposed to improve implementational compatibility between hexahedral and prismatic elements.

中文翻译：

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用于在棱柱元件上快速积分 DPG 矩阵的求和分解

摘要 高阶有限元 (FE) 方法在波传播等许多应用中具有显着优势，其中高阶形状函数有助于减轻污染（色散）误差。然而，高阶系统的经典组装在计算上很繁重，需要评估许多点正交方案。当采用不连续的 Petrov-Galerkin (DPG) FE 方法时，使用丰富的测试空间进一步增加了系统组装的计算负担，增加了改进组装技术的相关性。Sum factorization——一种利用形状函数的张量积结构来加速数值积分的技术——在参考文献中提出。[10] 用于在六面体元素上组装 DPG 矩阵，将计算复杂度从 O ( p 9 ) 阶降低到 O ( p 7 )（其中 p 表示多项式阶）。在这项工作中，我们通过将棱柱形函数表示为二维单纯形和一维区间形函数的张量积，将和因式分解的概念扩展到棱柱元上的 DPG 矩阵的构造。出乎意料的是，部分张量化的棱柱形函数的总和分解例程实现了与参考文献中提出的完全张量化的六面体形状函数（作为一维区间形状函数的乘积）的和分解相同的 O ( p 7 ) 复杂度。[10]。在整个工作中，我们坚持精确序列能量空间理论，为每个 3D FE 精确序列能量空间——H1、H(curl)、H(div) 和 L2 提出求和分解例程。给出了在每个精确序列能量空间中在棱柱元件上构建 DPG 格拉姆矩阵的计算结果，证实了预期的 O ( p 7 ) 复杂度。此外，考虑了为棱柱单元上的超弱麦克斯韦问题构建 DPG 系统，并提出了六面体单元的部分张量和分解，以提高六面体和棱柱单元之间的实现兼容性。