The greater arithmetic intensity of high‐order finite element discretizations makes them attractive for implementation on next‐generation hardware, but assembly of high‐order finite element operators as matrices is prohibitively expensive. As a result, the development of general algebraic solvers for such operators has been an open research challenge. Fast matrix‐free application of high‐order operators has received significant attention in the literature in the context of Poisson‐type problems, but preconditioners and solvers for inverting more general operators are not very well‐developed. In this paper, we consider the problem of preconditioning a definite Maxwell operator at high polynomial order without assembling a matrix. We show that given efficient preconditioners for high‐order H¹ finite element problems on the same mesh, efficient H(curl) preconditioners can be constructed in an auxiliary space framework. We demonstrate the resulting preconditioners in a practical setting with tensor‐product basis functions on an unstructured mesh of quadrilaterals. Our approach uses a sparsified H¹ solver constructed on a low‐order mesh of the nodal points of the underlying high‐order space, and we show that the resulting H(curl) preconditioner is effective at very high polynomial orders for two‐dimensional model problems with complicated geometry, varying piecewise constant coefficients, and curved elements. The resulting preconditioner scales with nearly optimal O(p^d + 1) floating point operation count and optimal O(p^d) memory transfer requirements, outperforming existing Maxwell preconditioners in the high‐order regime.

中文翻译：

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高阶H（curl）离散化的无矩阵预处理

高阶有限元离散化的较高算术强度使其对于在下一代硬件上实现具有吸引力，但是将高阶有限元算子作为矩阵进行组装的成本过高。因此，为此类算子开发通用代数求解器已成为一个开放的研究挑战。在泊松型问题的背景下，高阶算子的快速无矩阵应用受到了文献的广泛关注，但是用于反转更多一般算子的预处理器和求解器却没有得到很好的开发。在本文中，我们考虑了在不组装矩阵的情况下以高多项式对预定的Maxwell算子进行预处理的问题。我们证明了给定的高效预处理器对于高阶H ¹在同一网格上的有限元问题中，可以在辅助空间框架中构造有效的H（卷曲）预处理器。我们在非结构化四边形网格上使用张量乘积​​基函数在实际环境中演示了预处理器。我们的方法使用了稀疏的H ¹解算器，该解算器构造在基础高阶空间的节点的低阶网格上，并且我们证明了所得的H（卷曲）预处理器对于二维模型的非常高的多项式阶数有效几何形状复杂，分段常数系数变化以及弯曲元素的问题。最终的预处理器缩放比例接近于最佳O（p^{d  + 1}）浮点运算计数和最佳O（ p ^d）内存传输要求，在高阶状态下优于现有的Maxwell预处理器。