SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1429-B1461, January 2020.
Matrix-free finite element implementations for large applications provide an attractive alternative to standard sparse matrix data formats due to the significantly reduced memory consumption. Here, we show that they are also competitive with respect to the run-time in the low-order case if combined with suitable stencil scaling techniques. We focus on variable coefficient vector-valued partial differential equations as they arise in many physical applications. The presented method is based on scaling constant reference stencils originating from a linear finite element discretization instead of evaluating the bilinear forms on the fly. This method assumes the usage of hierarchical hybrid grids, and it may be applied to vector-valued second-order elliptic partial differential equations directly or as a part of more complicated problems. We provide theoretical and experimental performance estimates showing the advantages of this new approach compared to the traditional on-the-fly integration and stored matrix approaches. In our numerical experiments, we consider two specific mathematical models, namely, linear elastostatics and incompressible Stokes flow. The final example considers a nonlinear shear-thinning generalized Newtonian fluid. For this type of nonlinearity, we present an efficient approach for computing a regularized strain rate which is then used to define the nodewise viscosity. Depending on the compute architecture, we could observe maximum speedups of 64% and 122% compared to the on-the-fly integration. The largest considered example involved solving a Stokes problem with 12288 compute cores on the state-of-the-art supercomputer SuperMUC-NG.

中文翻译：

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混合网格上矢量值PDE的模板缩放及其在广义牛顿流体中的应用

SIAM科学计算杂志，第42卷，第6期，第B1429-B1461页，2020年1月。
由于显着减少了内存消耗，适用于大型应用程序的无矩阵有限元实现为标准稀疏矩阵数据格式提供了一种有吸引力的替代方法。在这里，我们表明，如果与适当的模板缩放技术相结合，它们在低阶情况下的运行时间也具有竞争力。我们关注可变系数矢量值偏微分方程，因为它们出现在许多物理应用中。提出的方法基于缩放源自线性有限元离散化的恒定参考模具，而不是动态评估双线性形式。该方法假定使用分层混合网格，并且可以直接应用于矢量值的二阶椭圆偏微分方程，也可以作为更复杂问题的一部分应用。我们提供了理论和实验性能评估，显示了这种新方法与传统的即时集成和存储矩阵方法相比的优势。在我们的数值实验中，我们考虑两个特定的数学模型，即线性弹性静力学和不可压缩的斯托克斯流。最后一个例子考虑了非线性剪切稀化的广义牛顿流体。对于这种类型的非线性，我们提出了一种有效的方法来计算规则应变率，然后将其用于定义节点粘度。根据计算架构的不同，与即时集成相比，我们可以观察到最大加速分别为64％和122％。考虑到的最大示例是在最新的超级计算机SuperMUC-NG上使用12288个计算内核解决斯托克斯问题。