Advertisement

Block-Preconditioners for Conforming and Non-conforming FEM Discretizations of the Cahn-Hilliard Equation

  • P. Boyanova
  • M. Do-Quang
  • M. Neytcheva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7116)

Abstract

We consider preconditioned iterative solution methods to solve the algebraic systems of equations arising from finite element discretizations of multiphase flow problems, based on the phase-field model.

The aim is to solve coupled physics problems, where both diffusive and convective processes take place simultaneously in time and space. To model the above, a coupled system of partial differential equations has to be solved, consisting of the Cahn-Hilliard equation to describe the diffusive interface and the time-dependent Navier-Stokes equation, to follow the evolution of the convection field in time.

We focus on the construction and efficiency of preconditioned iterative solution methods for the linear systems, arising after conforming and non-conforming finite element discretizations of the Cahn-Hilliard equation in space and implicit discretization schemes in time. The non-linearity of the phase-separation process is treated by Newton’s method. The resulting matrices admit a two-by-two block structure, utilized by the preconditioning techniques, proposed in the current work. We discuss approximation estimates of the preconditioners and include numerical experiments to illustrate their behaviour.

Keywords

Mass Matrix Iterative Solution Method Element Space Versus Complex Symmetric Linear System Pivot Block 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Axelsson, O.: Iterative Solution Methods. Cambridge University Press (1994)Google Scholar
  2. 2.
    Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Lin. Alg. Appl. 7, 197–218 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Axelsson, O., Neytcheva, M.: Operator splittings for solving nonlinear, coupled multiphysics problems with an application to the numerical solution of an interface problem, TR 2011-009, Institute for Information Technology, Uppsala Univ. (2011)Google Scholar
  4. 4.
    Bejanov, B., Guermond, J.-L., Minev, P.D.: A locally div-free projection scheme for incompressible flows based on non-conforming elements. Int. J. Numer. Meth. Fluids 49, 549–568 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Boyanova, P., Do-Quang, M., Neytcheva, M.: Solution methods for the Cahn-Hilliard equation discretized by conforming and non-conforming finite elements, TR 2011-004, Institute for Information Technology, Uppsala University (2011)Google Scholar
  6. 6.
    Do-Quang, M., Amberg, G.: The splash of a ball hitting a liquid surface: Numerical simulation of the influence of wetting. Journal Physic of Fluid (2008)Google Scholar
  7. 7.
    Elliott, C.M., French, D.A., Milner, F.A.: A second order splitting method for the Cahn-Hilliard Equation. Numer. Math. 54, 575–590 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Garcke, H.: On Cahn-Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh 133A, 307–331 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kaporin, I.E.: High Quality Preconditioning of a General Symmetric Positive Definite Matrix Based on its U T U + U T R + R T U-decomposition. Numer. Linear Algebra Appl. 5, 483–509 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Novick-Cohen, A.: The Cahn-Hilliard Equation: From Backwards Diffusion to Surface Diffusion, Draft. in preparation under contract with Cambridge University Press, http://www.math.technion.ac.il/~amync/
  11. 11.
    Pearson, J., Wathen, A.J.: A New Approximation of the Schur Complement in Preconditioners for PDE Constrained Optimization. The Mathematical institute, University of Oxford. Technical report, November 24 (2010), http://eprints.maths.ox.ac.uk/1021/
  12. 12.
    Villanueva, W., Amberg, G.: Some generic capillary-driven flows. International Journal of Multiphase Flow 32, 1072–1086 (2006)zbMATHCrossRefGoogle Scholar
  13. 13.
    Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA Journal of Numerical Analysis 7, 449–457 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Yue, P., Zhou, C., Feng, J.J., Ollivier-Gooch, C.F., Hu, H.H.: Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing. Journal of Computational Physics 219, 47–67 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • P. Boyanova
    • 1
    • 2
  • M. Do-Quang
    • 3
  • M. Neytcheva
    • 1
  1. 1.Uppsala UniversityUppsalaSweden
  2. 2.Institute for Information and Communication TechnologyBASSofiaBulgaria
  3. 3.Department of Mechanics, Linné Flow CentreRoyal Institute of TechnologyStockholmSweden

Personalised recommendations