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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)
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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 

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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