Convergence of Multi-level Algorithms for a Class of Nonlinear Problems
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Convergence of Multi-level Algorithms for a Class of Nonlinear Problems Dongho Kim1 · Eun-Jae Park2
· Boyoon Seo3
Received: 25 November 2019 / Revised: 16 July 2020 / Accepted: 18 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this article, we develop and analyze two-grid/multi-level algorithms via mesh refinement in the abstract framework of Brezzi, Rappaz, and Raviart for approximation of branches of nonsingular solutions. Optimal fine grid accuracy of two-grid/multi-level algorithms can be achieved via the proper scaling of relevant meshes. An important aspect of the proposed algorithms is the use of mesh refinement in conjunction with Newton-type methods for system solution in contrast to the usual Newton’s method on a fixed mesh. The pseudostressvelocity formulation of the stationary, incompressible Navier–Stokes equations is considered as an application and the Raviart–Thomas mixed finite element spaces are used for the approximation. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed. Keywords Nonlinear problems · Multi-level Mesh refinement · Error estimates · Two-grid algorithm · Pseudostress-velocity · Navier–Stokes equations Mathematics Subject Classification 65N15 · 65N30 · 65N50 · 76D05
Dongho Kim was supported by NRF-2018R1D1A1B0705058313. Eun-Jae Park was supported by the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT (NRF-2015R1A5A1009350 and NRF-2019R1A2C2090021). Boyoon Seo was supported by NRF-2020R1I1A1A0107036.
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Eun-Jae Park [email protected] Dongho Kim [email protected] Boyoon Seo [email protected]
1
University College, Yonsei University, Seoul 03722, Korea
2
Department of Computational Science and Engineering, Yonsei University, Seoul 03722, Korea
3
Department of Mathematics, Yonsei University, Seoul 03722, Korea 0123456789().: V,-vol
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Page 2 of 23
Journal of Scientific Computing
(2020) 84:34
1 Introduction Since the pioneering work of Xu [27,28], two-grid methods have received a lot of attention and successfully applied to approximate various (linear and) nonlinear problems (see, for example, [2,9,14,15,19,20,22,29]). The two-grid scheme is based on passing information between finite element equations defined on two grids of different mesh sizes. In the first step, a nonlinear problem itself is solved in a coarse space, i.e., finite dimensional space with coarse grids. In the second step, the nonlinear problem is linearized locally at the solution obtained in the coarse space. Then, the linearized problem is solved in a fine space. For better performance, this process can be iterated on a sequence of linearized problems with increasing dimensions. In this paper, we propose multi-level algorithms for a class of nonlinear problems using a two-grid idea. For error analysis, we use the abstract framework of Brezzi, Rappaz, and Raviart (BRR) [4] and the approach proposed by Caloz and Rappaz [7]. The f
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