Hybrid multigrid methods for high-order discontinuous Galerkin discretizations
Hybrid multigrid methods for high-order discontinuous Galerkin discretizations
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DOI：
10.1016/j.jcp.2020.109538
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
Niklas Fehn;Peter Munch;W. Wall;M. Kronbichler
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文献类型：
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作者：
Niklas Fehn;Peter Munch;W. Wall;M. Kronbichler
目前的工作开发的混合多重网格方法高阶不连续Galerkin离散椭圆问题，这是，例如，不可压缩流求解器在计算流体动力学领域的一个关键组成部分。张量积元素的快速矩阵运算被用来设计一个计算效率高的PDE求解器。多重网格层次结构利用几何，多项式和代数粗化的所有可能性，针对复杂几何形状的工程应用。此外，从不连续到连续函数空间的转移是在多重网格层次结构内进行的。这不仅进一步降低了粗网格问题的问题大小，而且导致最适合作为粗网格求解器应用的最先进的代数多重网格方法的离散化。相关的设计选择，关于选择最佳的多重网格粗化策略之间的各种可能性进行了讨论与度量的计算成本作为驱动力的算法选择。我们发现，以最高的多项式次数（或在最细的网格上）转移到连续函数空间，然后进行多项式和几何粗化，会显示出最佳的整体性能。这种特殊的多重网格策略的成功是由于一个显着减少迭代次数相比，从不连续的连续函数空间在最低的多项式次数（或粗网格）的转移。的粗化战略转移到一个连续的函数空间上的最好的水平导致一个多重网格算法，是强大的对称内部惩罚方法的惩罚参数。详细的数值研究进行了一系列的例子，从学术测试情况下，更复杂的，实际相关的几何形状。性能比较国家的最先进的方法从文献中证明了所提出的多重网格算法的多功能性和计算效率。
The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems, which are, for example, a key ingredient of incompressible flow solvers in the field of computational fluid dynamics. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE solver. The multigrid hierarchy exploits all possibilities of geometric, polynomial, and algebraic coarsening, targeting engineering applications on complex geometries. Additionally, a transfer from discontinuous to continuous function spaces is performed within the multigrid hierarchy. This does not only further reduce the problem size of the coarse-grid problem, but also leads to a discretization most suitable for state-of-the-art algebraic multigrid methods applied as coarse-grid solver. The relevant design choices regarding the selection of optimal multigrid coarsening strategies among the various possibilities are discussed with the metric of computational costs as the driving force for algorithmic selections. We find that a transfer to a continuous function space at highest polynomial degree (or on the finest mesh), followed by polynomial and geometric coarsening, shows the best overall performance. The success of this particular multigrid strategy is due to a significant reduction in iteration counts as compared to a transfer from discontinuous to continuous function spaces at lowest polynomial degree (or on the coarsest mesh). The coarsening strategy with transfer to a continuous function space on the finest level leads to a multigrid algorithm that is robust with respect to the penalty parameter of the symmetric interior penalty method. Detailed numerical investigations are conducted for a series of examples ranging from academic test cases to more complex, practically relevant geometries. Performance comparisons to state-of-the-art methods from the literature demonstrate the versatility and computational efficiency of the proposed multigrid algorithms.