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Erschienen in: Calcolo 4/2020

01.12.2020

Parallel iterative stabilized finite element methods based on the quadratic equal-order elements for incompressible flows

verfasst von: Bo Zheng, Yueqiang Shang

Erschienen in: Calcolo | Ausgabe 4/2020

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Abstract

Combining the quadratic equal-order stabilized method with the approach of local and parallel finite element computations and classical iterative methods for the discretization of the steady-state Navier–Stokes equations, three parallel iterative stabilized finite element methods based on fully overlapping domain decomposition are proposed and compared in this paper. In these methods, each processor independently computes an approximate solution in its own subdomain using a global composite mesh that is fine around its own subdomain and coarse elsewhere, making the methods be easy to implement based on existing codes and have low communication complexity. Under some (strong) uniqueness conditions, stability and convergence theory of the parallel iterative stabilized methods are derived. Numerical tests are also performed to demonstrate the stability, convergence orders and high efficiency of the proposed methods.
Literatur
1.
Zurück zum Zitat Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1984) MATH Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1984) MATH
2.
Zurück zum Zitat Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier–Stokes Equations. Springer, Berlin (1979) CrossRef Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier–Stokes Equations. Springer, Berlin (1979) CrossRef
3.
Zurück zum Zitat Girault, V., Raviart, P.A.: Finite Element Method for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1987) MATH Girault, V., Raviart, P.A.: Finite Element Method for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1987) MATH
4.
Zurück zum Zitat Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accomodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59, 85–99 (1986) CrossRef Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accomodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59, 85–99 (1986) CrossRef
5.
Zurück zum Zitat Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov–Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982) CrossRef Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov–Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982) CrossRef
6.
Zurück zum Zitat Hughes, T.J.R., Franca, L.P., Hulbert, G.M.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective–diffusive equations. Comput. Methods Appl. Mech. Eng. 73, 173–189 (1989) MathSciNetCrossRef Hughes, T.J.R., Franca, L.P., Hulbert, G.M.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective–diffusive equations. Comput. Methods Appl. Mech. Eng. 73, 173–189 (1989) MathSciNetCrossRef
7.
Zurück zum Zitat Masud, A., Khurram, R.A.: A multiscale finite element method for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 195, 1750–1777 (2006) MathSciNetCrossRef Masud, A., Khurram, R.A.: A multiscale finite element method for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 195, 1750–1777 (2006) MathSciNetCrossRef
8.
Zurück zum Zitat Blasco, J., Codina, R.: Stabilization finite elements method for the transient Navier–Stokes equations based on a pressure gradient projection. Comput. Methods Appl. Mech. Eng. 182, 277–300 (2000) CrossRef Blasco, J., Codina, R.: Stabilization finite elements method for the transient Navier–Stokes equations based on a pressure gradient projection. Comput. Methods Appl. Mech. Eng. 182, 277–300 (2000) CrossRef
9.
Zurück zum Zitat Li, J., He, Y.N.: A stabilized finite element method based on two local Gauss integrations for the Stokes equations. J. Comput. Appl. Math. 214, 58–65 (2008) MathSciNetCrossRef Li, J., He, Y.N.: A stabilized finite element method based on two local Gauss integrations for the Stokes equations. J. Comput. Appl. Math. 214, 58–65 (2008) MathSciNetCrossRef
10.
Zurück zum Zitat He, Y.N., Li, J.: A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations. Appl. Numer. Math. 58, 1503–1514 (2008) MathSciNetCrossRef He, Y.N., Li, J.: A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations. Appl. Numer. Math. 58, 1503–1514 (2008) MathSciNetCrossRef
11.
Zurück zum Zitat Zheng, H.B., Shan L., Hou, Y.R.: A quadratic equal-order stabilized method for Stokes problem based on two local Gauss integrations. Numer. Methods Partial Differ. Equ. 26, 180–1190 (2010) MathSciNetCrossRef Zheng, H.B., Shan L., Hou, Y.R.: A quadratic equal-order stabilized method for Stokes problem based on two local Gauss integrations. Numer. Methods Partial Differ. Equ. 26, 180–1190 (2010) MathSciNetCrossRef
12.
Zurück zum Zitat Huang, P.Z., Feng, X.L., Liu, D.M.: Two-level stabilized method based on three corrections for the stationary Navier–Stokes equations. Appl. Numer. Math. 62, 988–1001 (2012) MathSciNetCrossRef Huang, P.Z., Feng, X.L., Liu, D.M.: Two-level stabilized method based on three corrections for the stationary Navier–Stokes equations. Appl. Numer. Math. 62, 988–1001 (2012) MathSciNetCrossRef
13.
Zurück zum Zitat Qiu, H.L., An, R., Mei, L.Q., Xue, C.F.: Two-step algorithms for the stationary incompressible Navier–Stokes equations with friction boundary conditions. Appl. Numer. Math. 120, 97–114 (2017) MathSciNetCrossRef Qiu, H.L., An, R., Mei, L.Q., Xue, C.F.: Two-step algorithms for the stationary incompressible Navier–Stokes equations with friction boundary conditions. Appl. Numer. Math. 120, 97–114 (2017) MathSciNetCrossRef
14.
Zurück zum Zitat Zhang, G.L., Su, H.Y., Feng, X.L.: A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem. Numer. Heat Transf. B Fundam. 73, 329–341 (2018) CrossRef Zhang, G.L., Su, H.Y., Feng, X.L.: A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem. Numer. Heat Transf. B Fundam. 73, 329–341 (2018) CrossRef
15.
Zurück zum Zitat Zheng, B., Shang, Y.Q.: A parallel quadratic equal-order stabilized finite element algorithm based on fully overlapping domain decomposition for the steady Navier–Stokes equations, Submitted for publication Zheng, B., Shang, Y.Q.: A parallel quadratic equal-order stabilized finite element algorithm based on fully overlapping domain decomposition for the steady Navier–Stokes equations, Submitted for publication
16.
Zurück zum Zitat Zheng, B., Shang, Y.Q.: Parallel iterative stabilized finite element algorithms based on the lowest equal-order elements for the stationary Navier–Stokes equations. Appl. Math. Comput. 357, 35–56 (2019) MathSciNetMATH Zheng, B., Shang, Y.Q.: Parallel iterative stabilized finite element algorithms based on the lowest equal-order elements for the stationary Navier–Stokes equations. Appl. Math. Comput. 357, 35–56 (2019) MathSciNetMATH
17.
Zurück zum Zitat Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69, 881–909 (2000) MathSciNetCrossRef Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69, 881–909 (2000) MathSciNetCrossRef
18.
Zurück zum Zitat Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problem. Adv. Comput. Math. 14, 293–327 (2001) MathSciNetCrossRef Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problem. Adv. Comput. Math. 14, 293–327 (2001) MathSciNetCrossRef
19.
Zurück zum Zitat He, Y.N., Xu, J.C., Zhou, A.H., Li, J.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109, 415–434 (2008) MathSciNetCrossRef He, Y.N., Xu, J.C., Zhou, A.H., Li, J.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109, 415–434 (2008) MathSciNetCrossRef
20.
Zurück zum Zitat Shang, Y.Q., Wang, K.: Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations. Numer. Algorithms 54, 195–218 (2010) MathSciNetCrossRef Shang, Y.Q., Wang, K.: Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations. Numer. Algorithms 54, 195–218 (2010) MathSciNetCrossRef
21.
Zurück zum Zitat He, Y.N., Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for the Navier–Stokes problem. J. Comput. Math. 24, 227–238 (2006) MathSciNetMATH He, Y.N., Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for the Navier–Stokes problem. J. Comput. Math. 24, 227–238 (2006) MathSciNetMATH
22.
Zurück zum Zitat Shang, Y.Q., He, Y.N., Kim, D.W, Zhou, X.J.: A new parallel finite element algorithm for the stationary Navier–Stokes equations. Finite Elem. Anal. Des. 47, 1262–1279 (2011) MathSciNetCrossRef Shang, Y.Q., He, Y.N., Kim, D.W, Zhou, X.J.: A new parallel finite element algorithm for the stationary Navier–Stokes equations. Finite Elem. Anal. Des. 47, 1262–1279 (2011) MathSciNetCrossRef
23.
Zurück zum Zitat Shang, Y.Q., He, Y.N.: A parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 209, 172–183 (2012) MathSciNetCrossRef Shang, Y.Q., He, Y.N.: A parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 209, 172–183 (2012) MathSciNetCrossRef
24.
Zurück zum Zitat Du, G.Z., Hou, Y.R., Zuo, L.Y.: Local and parallel finite element method for the mixed Navier–Stokes/Darcy model. Int. J. Comput. Math. 93, 1155–1172 (2015) MathSciNetCrossRef Du, G.Z., Hou, Y.R., Zuo, L.Y.: Local and parallel finite element method for the mixed Navier–Stokes/Darcy model. Int. J. Comput. Math. 93, 1155–1172 (2015) MathSciNetCrossRef
25.
Zurück zum Zitat Zhang, Y.H., Hou, Y.R., Shan, L., Dong, X.J.: Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics. Numer. Methods Partial Differ. Equ. 33, 1513–1539 (2017) MathSciNetCrossRef Zhang, Y.H., Hou, Y.R., Shan, L., Dong, X.J.: Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics. Numer. Methods Partial Differ. Equ. 33, 1513–1539 (2017) MathSciNetCrossRef
26.
Zurück zum Zitat Tang, Q.L., Huang, Y.Q.: Local and parallel finite flement algorithm based on Oseen-type iteration for the stationary incompressible MHD flow. J. Sci. Comput. 70, 1–26 (2017) MathSciNetCrossRef Tang, Q.L., Huang, Y.Q.: Local and parallel finite flement algorithm based on Oseen-type iteration for the stationary incompressible MHD flow. J. Sci. Comput. 70, 1–26 (2017) MathSciNetCrossRef
27.
Zurück zum Zitat Yang, Y.D., Han, J.Y.: The multilevel mixed finite element discretizations based on local defect-correction for the Stokes eigenvalue problem. Comput. Meth. Appl. Mech. Eng. 289, 249–266 (2015) MathSciNetCrossRef Yang, Y.D., Han, J.Y.: The multilevel mixed finite element discretizations based on local defect-correction for the Stokes eigenvalue problem. Comput. Meth. Appl. Mech. Eng. 289, 249–266 (2015) MathSciNetCrossRef
28.
Zurück zum Zitat Bi, H., Han, J.Y., Yang, Y.D.: Local and parallel finite element algorithms for the transmission eigenvalue problem. J. Sci. Comput. 78, 351–375 (2019) MathSciNetCrossRef Bi, H., Han, J.Y., Yang, Y.D.: Local and parallel finite element algorithms for the transmission eigenvalue problem. J. Sci. Comput. 78, 351–375 (2019) MathSciNetCrossRef
29.
Zurück zum Zitat Zheng, H.B., Shi, F., Hou, Y.R., et al.: New local and parallel finite element algorithm based on the partition of unity. J. Math. Anal. Appl. 435, 1–19 (2016) MathSciNetCrossRef Zheng, H.B., Shi, F., Hou, Y.R., et al.: New local and parallel finite element algorithm based on the partition of unity. J. Math. Anal. Appl. 435, 1–19 (2016) MathSciNetCrossRef
30.
Zurück zum Zitat Du, G.Z., Zuo, L.Y.: A parallel partition of unity scheme based on two-grid discretizations for the Navier–Stokes problem. J. Sci. Comput. 4, 1–18 (2017) Du, G.Z., Zuo, L.Y.: A parallel partition of unity scheme based on two-grid discretizations for the Navier–Stokes problem. J. Sci. Comput. 4, 1–18 (2017)
31.
Zurück zum Zitat Shang, Y.Q.: A parallel subgrid stabilized finite element method based on fully overlapping domain decomposition for the Navier–Stokes equations. J. Math. Anal. Appl. 403, 667–679 (2013) MathSciNetCrossRef Shang, Y.Q.: A parallel subgrid stabilized finite element method based on fully overlapping domain decomposition for the Navier–Stokes equations. J. Math. Anal. Appl. 403, 667–679 (2013) MathSciNetCrossRef
32.
Zurück zum Zitat Shang, Y.Q.: Parallel defect-correction algorithms based on finite element discretization for the Navier–Stokes equations. Comput. Fluids 79, 200–212 (2013) MathSciNetCrossRef Shang, Y.Q.: Parallel defect-correction algorithms based on finite element discretization for the Navier–Stokes equations. Comput. Fluids 79, 200–212 (2013) MathSciNetCrossRef
33.
Zurück zum Zitat Shang, Y.Q., Qin, J.: Parallel finite element variational multiscale algorithms for incompressible flow at high Reyonds numbers. Appl. Numer. Math. 117, 1–21 (2017) MathSciNetCrossRef Shang, Y.Q., Qin, J.: Parallel finite element variational multiscale algorithms for incompressible flow at high Reyonds numbers. Appl. Numer. Math. 117, 1–21 (2017) MathSciNetCrossRef
34.
Zurück zum Zitat He, Y.N., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 1351–1359 (2009) MathSciNetCrossRef He, Y.N., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 1351–1359 (2009) MathSciNetCrossRef
35.
Zurück zum Zitat Shang, Y.Q., He, Y.N.: Parallel iterative finite element algorithms based on full domain decomposition for the stationary Navier–Stokes equations. Appl. Numer. Math. 60, 719–737 (2010) MathSciNetCrossRef Shang, Y.Q., He, Y.N.: Parallel iterative finite element algorithms based on full domain decomposition for the stationary Navier–Stokes equations. Appl. Numer. Math. 60, 719–737 (2010) MathSciNetCrossRef
36.
Zurück zum Zitat Adams, R.: Sobolev Spaces. Academaic Press Inc, New York (1975) MATH Adams, R.: Sobolev Spaces. Academaic Press Inc, New York (1975) MATH
37.
Zurück zum Zitat Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. Part I: regularity of solutions and second-order spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982) MathSciNetCrossRef Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. Part I: regularity of solutions and second-order spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982) MathSciNetCrossRef
38.
Zurück zum Zitat Kechkar, N., Silvester, D.: Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58, 1–10 (1992) MathSciNetCrossRef Kechkar, N., Silvester, D.: Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58, 1–10 (1992) MathSciNetCrossRef
39.
40.
Zurück zum Zitat Arioli, M., Loghin, D., Wathen, A.J.: Stopping criteria for iterations in finite element methods. Numer. Math. 99(3), 381–410 (2005) MathSciNetCrossRef Arioli, M., Loghin, D., Wathen, A.J.: Stopping criteria for iterations in finite element methods. Numer. Math. 99(3), 381–410 (2005) MathSciNetCrossRef
Metadaten
Titel
Parallel iterative stabilized finite element methods based on the quadratic equal-order elements for incompressible flows
verfasst von
Bo Zheng
Yueqiang Shang
Publikationsdatum
01.12.2020
Verlag
Springer International Publishing
Erschienen in
Calcolo / Ausgabe 4/2020
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-020-00382-6

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