A Parallel Iterative Finite Element Method for the Linear Elliptic Equations
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A Parallel Iterative Finite Element Method for the Linear Elliptic Equations Guangzhi Du1 · Liyun Zuo2 Received: 24 January 2018 / Revised: 25 February 2020 / Accepted: 13 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract By combining the two-grid discretization with the partition of unity method, a parallel iterative finite element method for the linear elliptic equations is proposed and investigated. Since the construction of the partition of unity is based on the coarse mesh triangulation, the computational domain of each subproblem can be divided automatically and the number of subproblems can be arbitrarily huge as the coarse mesh parameter H tends to zero. That means our method can be easily implemented in high performance supercomputers or cluster of workstations. Theoretical results based on a priori error estimation of the scheme are obtained, which indicate that our method can reach the optimal convergence orders within a few two-grid iterations. Numerical results are reported to assess the theoretical results. Keywords Finite element method · Partition of unity method · Parallel algorithm · Two-grid iterations Mathematics Subject Classification 65N15 · 65N30 · 65N55
1 Introduction Recently, due to the limits in computational resources, the development of efficient decoupling methods such as two-grid/two-level post-processing schemes and domain decomposition methods [4,14,15,20,21] for PDEs with high resolution has been an active research topic. By the observation that, for a solution to the PDEs, low frequency components can be approximated well on the coarse grid while high frequency components can be computed on
Subsidized by NSFC (Grant No. 11701343) and partially supported by NSFC (Grant No. 11801332), Subsidized by the Provincial Natural Science Foundation of Shandong (Grant No. ZR2017BA027).
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Guangzhi Du [email protected] Liyun Zuo [email protected]
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School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
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School of Mathematical Sciences, University of Jinan, Jinan 250022, China 0123456789().: V,-vol
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a fine grid by some local and parallel procedures, Xu and Zhou proposed some local and parallel approachs [27,28] for a class of linear and nonlinear elliptic boundary value problems. Following the ideas of Xu and Zhou for the local finite element discretization, some new local and parallel algorithms have been proposed for the steady Stokes equations [12,23], for the Navier–Stokes equations [10,11,24], for the Navier–Stokes/Darcy problems [6,31], and for other unsteady problems [5,19,25]. The most attractive feature of these algorithms is that the series of subproblems are independent once a coarse grid approximation is known and therefore it is a highly parallelized algorithm. However, there are still some ways to improve these local and parallel algorithms, such as, implementing domain decomposition in a better way, accomplishing the parallel proce
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