A two-scale approach for efficient on-the-fly operator assembly in\n massively parallel high performance multigrid codes Article Swipe
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· 2016
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.1608.06473
· OA: W4302883641
Matrix-free finite element implementations of massively parallel geometric\nmultigrid save memory and are often significantly faster than implementations\nusing classical sparse matrix techniques. They are especially well suited for\nhierarchical hybrid grids on polyhedral domains. In the case of constant\ncoefficients all fine grid node stencils in the interior of a coarse macro\nelement are equal. However, for non-polyhedral domains the situation changes.\nThen even for the Laplace operator, the non-linear element mapping leads to\nfine grid stencils that can vary from grid point to grid point. This\nobservation motivates a new two-scale approach that exploits a piecewise\npolynomial approximation of the fine grid operator with respect to the coarse\nmesh size. The low-cost evaluation of these surrogate polynomials results in an\nefficient stencil assembly on-the-fly for non-polyhedral domains that can be\nsignificantly more efficient than matrix-free techniques that are based on an\nelement-wise assembly. The performance analysis and additional hardware-aware\ncode optimizations are based on the Execution-Cache-Memory model. Several\naspects such as two-scale a priori error bounds and double discretization\ntechniques are presented. Weak and strong scaling results illustrate the\nbenefits of the new technique when used within large scale PDE solvers.\n
