Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 11, 2020

P1-nonconforming divergence-free finite element method on square meshes for Stokes equations

  • Chunjae Park EMAIL logo

Abstract

Recently, the P1-nonconforming finite element space over square meshes has been proved stable to solve Stokes equations with the piecewise constant space for velocity and pressure, respectively. In this paper, we will introduce its locally divergence-free subspace to solve the elliptic problem for the velocity only decoupled from the Stokes equation. The concerning system of linear equations is much smaller compared to the Stokes equations. Furthermore, it is split into two smaller ones. After solving the velocity first, the pressure in the Stokes problem can be obtained by an explicit method very rapidly.

JEL Classification: 65N30

Award Identifier / Grant number: NRF-2017R1D1A1A02019336

Funding statement: The present research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1A02019336).

References

[1] R. Altmann and C. Carstensen, P1-nonconforming finite elements on triangulations into triangles and quadrilaterals, SIAM J. Numer. Anal. 50 (2012), 418–438.10.1137/110823675Search in Google Scholar

[2] T. M. Austin, T. A. Manteuffel, and S. McCormick, A robust multilevel approach for minimizing H(div)-dominated functionals in an H1-conforming finite element space, Numer. Linear Algebra Appl. 11 (2004), 115–140.10.1002/nla.373Search in Google Scholar

[3] S. Brenner, F. Li, and L. Sung, A locally divergence-free nonconforming finite element method for the reduced time-harmonic Maxwell equations, Math. Comp. 76 (2007), 573–595.10.1090/S0025-5718-06-01950-8Search in Google Scholar

[4] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods Springer-Verlag, New York, 1991.10.1007/978-1-4612-3172-1Search in Google Scholar

[5] A. Buffa, C. de Falco, and G. Sangalli, IsoGeometric Analysis: stable elements for the 2D Stokes equation, Int. J. Numer. Methods Fluids 65 (2011), 1407–1422.10.1002/fld.2337Search in Google Scholar

[6] B. Cockburn, F. Li, and C. Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys. 194 (2004), 588–610.10.1016/j.jcp.2003.09.007Search in Google Scholar

[7] B. Cockburn, G. Kanschat, and D. Schötzau, A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations, J. Sci. Comput. 31 (2007), 61–73.10.1007/s10915-006-9107-7Search in Google Scholar

[8] J. A. Evans and T. J. R. Hughes, Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations, Math. Models Methods Appl. Sci. 23 (2013), 1421–1478.10.21236/ADA560496Search in Google Scholar

[9] V. Girault and P. A. Raviart, Finite element methods for the Navier–Stokes equations: Theory and Algorithms Springer-Verlag, New York, 1986.10.1007/978-3-642-61623-5Search in Google Scholar

[10] F. Hecht, Construction d’une base de fonctions P1 non conforme á divergence nulle dans ℝ3, RAIRO Anal. Numér. 15 (1981), 119–150.10.1051/m2an/1981150201191Search in Google Scholar

[11] R. Horn and C. Johnson, Matrix Analysis Cambridge University Press, New York, 1985.10.1017/CBO9780511810817Search in Google Scholar

[12] Y. Huang and S. Zhang, A lowest order divergence-free finite element on rectangular grids, Front. Math. China 6 (2011), 253–270.10.1007/s11464-011-0094-0Search in Google Scholar

[13] S. Kim, J. Yim, and D. Sheen, Stable cheapest nonconforming finite elements for the Stokes equations, J. Comp. Appl. Math. 299 (2016), 2–14.10.1016/j.cam.2015.06.021Search in Google Scholar

[14] C. Park, A Study on Locking phenomena in finite element methods, Ph.D. thesis Department of Mathematics, Seoul National University, Seoul, Korea, 2002.Search in Google Scholar

[15] C. Park and D. Sheen, P1-nonconforming quadrilateral finite element methods for second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), 624–640.10.1137/S0036142902404923Search in Google Scholar

[16] C. Park, Error analysis of Q1 − P0 for Stokes equations, (in preparation).Search in Google Scholar

[17] O. Pironneau, Finite Element Methods for Fluids Wiley, Chichester, 1989.Search in Google Scholar

[18] M. Powell and M. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3 (1977), 316–325.10.1145/355759.355761Search in Google Scholar

[19] S. K. Stein, Mathematics: The Man-Made Universe Dover Publications, New York, 1999.Search in Google Scholar

[20] F. Thomasset, Implementation of Finite Element Methods for Navier–Stokes Equations Springer-Verlag, New York–Berlin, 1981.10.1007/978-3-642-87047-7Search in Google Scholar

[21] C. A. Hall and X. Ye, Construction of null bases for the divergence operator associated with incompressible Navier–Stokes equations, Linear Algebra Appl. 171 (1992), 9–52.10.1016/0024-3795(92)90248-9Search in Google Scholar

[22] X. Ye and C. A. Hall, Discrete divergence-free basis for finite element methods, Numer. Algorithms 16 (1997), 365–380.10.1023/A:1019159702198Search in Google Scholar

[23] S. Zhang, A family of Qk+1,k × Qkk+1 divergence-free finite elements on rectangular grids, SIAM J. Numer. Anal. 47 (2009), 2090–2107.10.1137/080728949Search in Google Scholar

Received: 2019-05-08
Revised: 2019-09-04
Accepted: 2019-09-10
Published Online: 2020-12-11
Published in Print: 2020-12-16

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/jnma-2019-0056/html
Scroll to top button