Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T05:15:14.541Z Has data issue: false hasContentIssue false
  • English
  • Français

An AMG Preconditioner for Solving the Navier-Stokes Equations with a Moving Mesh Finite Element Method

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  19 October 2016

Yirong Wu  and
Heyu Wang
Yirong Wu*
Affiliation:
School of Mathematical Science, ZheJiang University, HangZhou, 310027, China
Heyu Wang*
Affiliation:
School of Mathematical Science, ZheJiang University, HangZhou, 310027, China
*
*Corresponding author. Email addresses:[email protected] Y. Wu), [email protected] (H. Wang)
*Corresponding author. Email addresses:[email protected] Y. Wu), [email protected] (H. Wang)
Get access

Abstract

AMG preconditioners are typically designed for partial differential equation solvers and divergence-interpolation in a moving mesh strategy. Here we introduce an AMG preconditioner to solve the unsteady Navier-Stokes equations by a moving mesh finite element method. A 4P1 – P1 element pair is selected based on the data structure of the hierarchy geometry tree and two-layer nested meshes in the velocity and pressure. Numerical experiments show the efficiency of our approach.

Keywords

Navier-Stokes systemalgebraic multigrid preconditionmoving mesh

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 4 , November 2016 , pp. 353 - 366
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Winslow, A.M., Numerical solution of the quasilinear Poisson equation in a nonuniform triangle mesh, J. Comput. Phys. 135, 128–138 (1966).Google Scholar
[2] Dvinsky, A.S., Adaptive grid generation from harmonic maps on Riemannian manifolds, J. Comput. Phys. 95, 450–476 (1991).Google Scholar
[3] Li, R., Tang, T., Zhang, P.W., Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys. 170, 562–588 (2001).CrossRefGoogle Scholar
[4] Di, Y., Li, R., Tang, T., and Zhang, P., Moving mesh finite element methods for the incompressible Navier-Stokes equations, SIAM J. Sci. Comput. 26, 1036–1056 (2005).CrossRefGoogle Scholar
[5] Wu, Y.R. and Wang, H.Y., Moving mesh finite element method for unsteady Navier-Stokes flow, East Asian J. Appl. Math. to appear.Google Scholar
[6] Bercovier, M. andPironneau, O., Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math. 33, 211–224 (1979).CrossRefGoogle Scholar
[7] Shen, L. and Xu, J.C., On a Schur complement operator arisen from Navier-Stokes equations and its preconditioning, Lecture Notes in Pure and Appl. Math. 202, 481–490 (1999).Google Scholar
[8] Xu, J.C., Iterative methods by space decomposition and subspace correction, SIAM Rev. 34, 581–613 (1992).Google Scholar
[9] He, Y.N., Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal. 41, 1263–1285 (2003).Google Scholar
[10] Benzi, M., Golub, G.H., and Liesen, J., Numerical solution of saddle point problems, Acta Numer. 14, 1–137 (2005).Google Scholar
[11] Bai, Z.Z. and Ng, M.K., On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput. 26, 1710–1724 (2005).CrossRefGoogle Scholar
[12] Bai, Z.Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comp. 75, 791–815 (2006).CrossRefGoogle Scholar
[13] Elman, H., Howle, V.E., Shadid, J., Silvester, D., and Tuminaro, R., Least squares preconditioners for stabilised discretisations of the Navier-Stokes equations, SIAM J. Sci. Comput. 30, 290–311 (2007).Google Scholar
[14] Elman, H.C. and Tuminaro, R., Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations, Electron. Trans. Numer. Anal. 35, 257–280 (2009).Google Scholar
[15] Benzi, M. and Olshanskii, M.A., An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28, 2095–2113 (2006).Google Scholar
[16] Benzi, M., Ng, M.K., Niu, Q., and Wang, Z., A relaxed dimensional factorisation preconditioner for the incompressible Navier-Stokes equations, J. Comput. Phys. 230, 6185–6202 (2011).Google Scholar
[17] Boyle, J., Mihajlovic, M.D., and Scott, J.A., Hsl_mi20: An efficient amg preconditioner for finite element problems in 3d, Internat. J. Numer. Methods Engrg. 82, 64–98 (2010).Google Scholar
[18] Elman, H.C., Silvester, D.J., and Wathen, A.J., Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford (2014).Google Scholar
[19] Li, R., On multi-mesh h-adaptive methods, J. Sci. Comput. 24, 321–341 (2005).Google Scholar
[20] Elman, H., Mihajlovi, M., and Silvester, D., Fast iterative solvers for buoyancy driven flow problems, J. Comput. Phys. 230, 3900–3914 (2011).Google Scholar
[21] Li, R., Tang, T., and Zhang, P.W.. A moving mesh finite element algorithm for singular problems in two and three space dimensions, J. Comput. Phys. 177, 365–393 (2002).Google Scholar
[22] Cao, W.M., Huang, W.Z., and Russell, R.D., An r-adaptive finite element method based upon moving mesh pdes, J. Comput. Phys. 149, 221–244 (1999).Google Scholar
[23] Dyke, M.V., An Album of Fluid Motion, Parabolic Press, Stanford (1982).Google Scholar