Advertisement

Domain Decomposition Approaches for PDE Based Mesh Generation

  • Ronald D. HaynesEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)

Abstract

Adaptive, partial differential equation (PDE) based, mesh generators are introduced. The mesh PDE is typically coupled to the physical PDE of interest and one has to be careful not to introduce undue computational burden. Here we provide an overview of domain decomposition approaches to reduce this computational overhead and provide a parallel solver for the coupled PDEs. A preview of a new analysis for optimized Schwarz methods for the mesh generation problem using the theory of M-functions is given. We conclude by introducing a two-grid method with FAS correction for the grid generation problem.

Notes

Acknowledgements

I would like to thank my former students Alex Howse, Devin Grant, and Abu Sarker for their assistance and some of the plots included in this paper, and also Felix Kwok for several discussions related to this work.

References

  1. 1.
    J.A. Acebrón, M.P. Busico, P. Lanucara, R. Spigler, Domain decomposition solution of elliptic boundary-value problems via Monte Carlo and quasi-Monte Carlo methods. SIAM J. Sci. Comput. 27(2), 440–457 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    D.A. Anderson, Equidistribution schemes, Poisson generators, and adaptive grids. Appl. Math. Comput. 24(3), 211–227 (1987)MathSciNetzbMATHGoogle Scholar
  3. 3.
    A. Bihlo, R.D. Haynes, Parallel stochastic methods for PDE based grid generation. Comput. Math. Appl. 68(8), 804–820 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Brandt, Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31(138), 333–390 (1977)MathSciNetCrossRefGoogle Scholar
  5. 5.
    W. Cao, W. Huang, R.D. Russell, A study of monitor functions for two-dimensional adaptive mesh generation. SIAM J. Sci. Comput. 20(6), 1978–1994 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    W.P. Crowley, An equipotential zoner on a quadrilateral mesh. Memo, Lawrence Livermore National Lab, 5 (1962)Google Scholar
  7. 7.
    C. de Boor, Good approximation by splines with variable knots, in Spline Functions and Approximation Theory, vol. 21, chap. 3. International Series of Numerical Mathematics, vol. 21 (Springer, Berlin, 1973), pp. 57–72Google Scholar
  8. 8.
    A.S. Dvinsky, Adaptive grid generation from harmonic maps on Riemannian manifolds. J. Comput. Phys. 95(2), 450–476 (1991)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M.J. Gander, R.D. Haynes, Domain decomposition approaches for mesh generation via the equidistribution principle. SIAM J. Numer. Anal. 50(4), 2111–2135 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S.K. Godunov, G.P. Prokopov, The use of moving meshes in gas-dynamical computations. USSR Comput. Math. Math. Phys. 12(2), 182–195 (1972)CrossRefGoogle Scholar
  11. 11.
    D. Grant, Acceleration techniques for mesh generation via domain decomposition methods. B.Sc., Memorial University of Newfoundland, St. John’s, Newfoundland (2015)Google Scholar
  12. 12.
    R.D. Haynes, A.J.M. Howse, Alternating Schwarz methods for partial differential equation-based mesh generation. Int. J. Comput. Math. 92(2), 349–376 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    R.D. Haynes, F. Kwok, Discrete analysis of domain decomposition approaches for mesh generation via the equidistribution principle. Math. Comput. 86(303), 233–273 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    R.D. Haynes, R.D. Russell, A Schwarz waveform moving mesh method. SIAM J. Sci. Comput. 29(2), 656–673 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    W. Huang, R.D. Russell, Adaptive Moving Mesh Methods. Applied Mathematical Sciences, vol. 174 (Springer, New York, 2011)Google Scholar
  16. 16.
    W. Huang, D.M. Sloan, A simple adaptive grid method in two dimensions. SIAM J. Sci. Comput. 15(4), 776–797 (1994)MathSciNetCrossRefGoogle Scholar
  17. 17.
    R.B. Kellogg, A nonlinear alternating direction method. Math. Comput. 23, 23–27 (1969)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (SIAM, Philadelphia, 2000)CrossRefGoogle Scholar
  19. 19.
    E. Peirano, D. Talay, Domain decomposition by stochastic methods, in Domain Decomposition Methods in Science and Engineering (National Autonomous University, México, 2003), pp. 131–147Google Scholar
  20. 20.
    W. Rheinboldt, On classes of n-dimensional nonlinear mappings generalizing several types of matrices, in Numerical Solution of Partial Differential Equations - II, ed. by B. Hubbard. Synspade 1970 (Academic, New York-London, 1971), pp. 501–545Google Scholar
  21. 21.
    W.C. Rheinboldt, On M-functions and their application to nonlinear Gauss-Seidel iterations and to network flows. J. Math. Anal. Appl. 32(2), 274–307 (1970)MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Sarker, Optimized Schwarz domain decomposition approaches for the generation of equidistributing grids. M.Sc., Memorial University of Newfoundland, St. John’s, Newfoundland (2015)Google Scholar
  23. 23.
    S. Schechter, Iteration methods for nonlinear problems. Trans. Am. Math. Soc. 104(1), 179–189 (1962)MathSciNetCrossRefGoogle Scholar
  24. 24.
    R. Spigler, A probabilistic approach to the solution of PDE problems via domain decomposition methods, in The Second International Conference on Industrial and Applied Mathematics, ICIAM (1991)Google Scholar
  25. 25.
    J.F. Thompson, F.C. Thames, C.W. Mastin, Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies. J. Comput. Phys. 15(3), 299–319 (1974)CrossRefGoogle Scholar
  26. 26.
    A.M. Winslow, Adaptive-mesh zoning by the equipotential method. Technical Report, Lawrence Livermore National Laboratory, CA (1981)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Memorial UniversitySt. John’sCanada

Personalised recommendations