标题： 快速无矩阵求解不连续伽辽金有限元算子 显示英文标题

标题： Fast matrix-free evaluation of discontinuous Galerkin finite element operators

摘要： 我们提出了一种算法框架，用于在四边形和六面体网格上基于求和分解的无矩阵不连续伽辽金有限元算子评估。 我们识别了一组用于单元和面上快速积分的内核，针对源自线性和非线性偏微分方程的广泛弱形式。 在深入的性能分析中比较了实现算子评估的不同算法和数据结构。 通过在多个单元和面上的向量化以及一维计算内核的奇偶分解，对求和分解内核进行了优化。 单独来看，我们的实现可以在英特尔Haswell和Broadwell处理器上达到高达60%的浮点运算峰值，在英特尔Knights Landing上达到高达50%的浮点运算峰值。 由于加载输入和输出向量、MPI幽灵交换以及处理变量系数和几何信息的内存带宽限制，完整的算子评估的吞吐量仅为上述值的一半左右。 我们的性能分析表明，对于所提出的实现，结果通常在可用内存带宽的10%以内，除了笛卡尔网格情况，其中收集操作和MPI通信的成本更为显著。 显示英文摘要

摘要： We present an algorithmic framework for matrix-free evaluation of discontinuous Galerkin finite element operators based on sum factorization on quadrilateral and hexahedral meshes. We identify a set of kernels for fast quadrature on cells and faces targeting a wide class of weak forms originating from linear and nonlinear partial differential equations. Different algorithms and data structures for the implementation of operator evaluation are compared in an in-depth performance analysis. The sum factorization kernels are optimized by vectorization over several cells and faces and an even-odd decomposition of the one-dimensional compute kernels. In isolation our implementation then reaches up to 60\% of arithmetic peak on Intel Haswell and Broadwell processors and up to 50\% of arithmetic peak on Intel Knights Landing. The full operator evaluation reaches only about half that throughput due to memory bandwidth limitations from loading the input and output vectors, MPI ghost exchange, as well as handling variable coefficients and the geometry. Our performance analysis shows that the results are often within 10\% of the available memory bandwidth for the proposed implementation, with the exception of the Cartesian mesh case where the cost of gather operations and MPI communication are more substantial.