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Simple Finite Element Numerical Simulation of Incompressible Flow Over Non-rectangular Domains and the Super-Convergence Analysis

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Abstract

In this paper, we apply a simple finite element numerical scheme, proposed in an earlier work (Liu in Math Comput 70(234):579–593, 2000), to perform a high resolution numerical simulation of incompressible flow over an irregular domain and analyze its boundary layer separation. Compared with many classical finite element fluid solvers, this numerical method avoids a Stokes solver, and only two Poisson-like equations need to be solved at each time step/stage. In addition, its combination with the fully explicit fourth order Runge–Kutta (RK4) time discretization enables us to compute high Reynolds number flow in a very efficient way. As an application of this robust numerical solver, the dynamical mechanism of the boundary layer separation for a triangular cavity flow with Reynolds numbers \(Re=10^4\) and \(Re=10^5\), including the precise values of bifurcation location and critical time, are reported in this paper. In addition, we provide a super-convergence analysis for the simple finite element numerical scheme, using linear elements over a uniform triangulation with right triangles.

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Acknowledgments

The authors greatly appreciate many helpful discussions with Wenbin Chen, Sigal Gottlieb, Yuan Liu and Chi-Wang Shu, in particular for their insightful suggestion and comments. This work is supported in part by the NSF DMS-1115420 NSF DMS-1418689 (C. Wang), NSFC 11271281 (C. Wang), NSFC 11171168 (Y. Xue) and the fund by China Scholarship Council (Y. Xue). Y. Xue thanks University of Massachusetts Dartmouth, for support during his visit. C. Wang also thanks Shanghai Center for Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, for support during his visit.

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Authors and Affiliations

  1. School of Mathematics Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Yunhua Xue

  2. Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA, 02747-2300, USA

    Cheng Wang

  3. Department of Physics and Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Jian-Guo Liu

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  1. Yunhua Xue
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  2. Cheng Wang
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  3. Jian-Guo Liu
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Correspondence to Cheng Wang.

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Xue, Y., Wang, C. & Liu, JG. Simple Finite Element Numerical Simulation of Incompressible Flow Over Non-rectangular Domains and the Super-Convergence Analysis. J Sci Comput 65, 1189–1216 (2015). https://doi.org/10.1007/s10915-015-0005-8

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