SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page C312-C334, January 2021.
Discontinuous Galerkin (dG) methods on meshes consisting of polygonal/polyhedral (henceforth, collectively termed as polytopic) elements have received considerable attention in recent years. Due to the physical frame basis functions used typically and the quadrature challenges involved, the matrix-assembly step for these methods is often computationally cumbersome. To address this important practical issue, this work proposes two parallel assembly implementation algorithms on Compute Unified Device Architecture--enabled graphics cards for the interior penalty dG method on polytopic meshes for various classes of linear PDE problems. We are concerned with both single graphics processing unit (GPU) parallelization, as well as with implementation on distributed GPU nodes. The results included showcase almost linear scalability of the quadrature step with respect to the number of GPU cores used since no communication is needed for the assembly step. In turn, this can justify the claim that polytopic dG methods can be implemented extremely efficiently, as any assembly computing time overhead compared to finite elements on “standard” simplicial or box-type meshes can be effectively circumvented by the proposed algorithms.

中文翻译：

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多面体网格上的 GPU 加速不连续伽辽金方法

SIAM 科学计算杂志，第 43 卷，第 4 期，第 C312-C334 页，2021 年 1 月。
近年来，由多边形/多面体（以下统称为多面体）元素组成的网格上的不连续伽辽金 (dG) 方法受到了相当多的关注。由于通常使用的物理框架基函数和所涉及的正交挑战，这些方法的矩阵组装步骤通常在计算上很麻烦。为了解决这个重要的实际问题，这项工作在计算统一设备架构上提出了两种并行装配实现算法——支持图形卡，用于针对各类线性 PDE 问题的多面网格上的内部惩罚 dG 方法。我们关注单个图形处理单元 (GPU) 并行化，以及分布式 GPU 节点上的实现。包括的结果展示了正交步骤相对于所用 GPU 内核数量的几乎线性可扩展性，因为组装步骤不需要通信。反过来，这可以证明多面体 dG 方法可以非常有效地实现的说法是正确的，因为与“标准”单纯形或盒型网格上的有限元相比，所提出的算法可以有效地规避任何装配计算时间开销。