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

  • Yirong Wu (a1) and Heyu Wang (a1)

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.

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*Corresponding author. Email addresses: Y. Wu), (H. Wang)
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[1] Winslow A.M., Numerical solution of the quasilinear Poisson equation in a nonuniform triangle mesh, J. Comput. Phys. 135, 128138 (1966).
[2] Dvinsky A.S., Adaptive grid generation from harmonic maps on Riemannian manifolds, J. Comput. Phys. 95, 450476 (1991).
[3] Li R., Tang T., Zhang P.W., Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys. 170, 562588 (2001).
[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, 10361056 (2005).
[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.
[6] Bercovier M. andPironneau O., Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math. 33, 211224 (1979).
[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, 481490 (1999).
[8] Xu J.C., Iterative methods by space decomposition and subspace correction, SIAM Rev. 34, 581613 (1992).
[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, 12631285 (2003).
[10] Benzi M., Golub G.H., and Liesen J., Numerical solution of saddle point problems, Acta Numer. 14, 1137 (2005).
[11] Bai Z.Z. and Ng M.K., On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput. 26, 17101724 (2005).
[12] Bai Z.Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comp. 75, 791815 (2006).
[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, 290311 (2007).
[14] Elman H.C. and Tuminaro R., Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations, Electron. Trans. Numer. Anal. 35, 257280 (2009).
[15] Benzi M. and Olshanskii M.A., An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28, 20952113 (2006).
[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, 61856202 (2011).
[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, 6498 (2010).
[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).
[19] Li R., On multi-mesh h-adaptive methods, J. Sci. Comput. 24, 321341 (2005).
[20] Elman H., Mihajlovi M., and Silvester D., Fast iterative solvers for buoyancy driven flow problems, J. Comput. Phys. 230, 39003914 (2011).
[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, 365393 (2002).
[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, 221244 (1999).
[23] Dyke M.V., An Album of Fluid Motion, Parabolic Press, Stanford (1982).
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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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