Skip to main content
SpringerLink
Log in

Optimized Schwarz Methods with Elliptical Domain Decompositions

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Over the past decade, partial differential equation models in elliptical geometries have become a focus of interest in several scientific and engineering applications: the classical studies of flow past a cylinder, the spherical particles in nano-fluids and spherical water filled domains are replaced by elliptical geometries which more accurately describe a wider class of physical problems of interest. Optimized Schwarz methods (OSMs) are among the best parallel methods for such models. We study here for the first time OSMs with elliptical domain decompositions, i.e. decompositions into an ellipse and elliptical rings. Using the technique of separation of variables, we decouple the spatial variables and reduce the subdomain problems to radial Mathieu like equations defined on finite intervals, which allows us to derive and study a new family of OSMs. Our analysis reveals that the optimized transmission parameters are not constants any more along the elliptical interfaces. We can prove however also that using the constant optimized parameters from the straight interface analysis in the literature scaled locally by the interface curvature is still efficient in an asymptotic sense, which leads to the important discovery of a unique factor in the optimized parameters and asymptotic performance determined by the geometry of the decomposition. We use numerical examples to illustrate our analysis and findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Availability of Data and Material

Not applicable.

Notes

  1. Here we neglect the fact that when the ellipse is angular-wise uniformly meshed with mesh size \(h_{\theta }\), the interface is not correspondingly uniformly meshed anymore.

References

  1. Amrein, W.O., Hinz, A.M., Pearson, D.B.: Sturm-Liouville Theory: Past and Present. Birkhäuser Verlag (2005)

  2. Baeva, M., Baev, P., Kaplan, A.: An analysis of the heat transfer from a moving elliptical cylinder. J. Phys. D Appl. Phys. 30(8), 1190 (1997)

    Article  Google Scholar 

  3. Bakker, M.: Modeling groundwater flow to elliptical lakes and through multi-aquifer elliptical inhomogeneities. Adv. Water Resour. 27(5), 497–506 (2004)

    Article  Google Scholar 

  4. Bakker, M., Kuhlman, K.L.: Computational issues and applications of line-elements to model subsurface flow governed by the modified Helmholtz equation. Adv. Water Resour. 34(9), 1186–1194 (2011)

    Article  Google Scholar 

  5. Bennequin, D., Gander, M.J., Gouarin, L., Halpern, L.: Optimized Schwarz waveform relaxation for advection reaction diffusion equations in two dimensions. Numer. Math. 134(3), 513–567 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bennequin, D., Gander, M.J., Halpern, L.: A homographic best approximation problem with application to optimized Schwarz waveform relaxation. Math. Comput. 78(265), 185–223 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. El Bouajaji, M., Dolean, V., Gander, M.J., Lanteri, S.: Optimized Schwarz methods for the time-harmonic Maxwell equations with damping. SIAM J. Sci. Comput. 34(4), A2048–A2071 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brimacombe, C., Corless, R.M., Zamir, M.: Computation and applications of Mathieu functions: a historical perspective. (2020). arXiv preprint arXiv:2008.01812

  9. Chaplygin, S.A.: New methods in the approximate integration of differential equations. Gostiekhizdat, Moscowa (1950). (in Russian)

    Google Scholar 

  10. Ciaramella, G., Gander, M.J.: Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains: Part II. SIAM J. Numer. Anal. 56(3), 1498–1524 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciaramella, G., Gander, M.J.: Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains: Part III. Electron. Trans. Numer. Anal. 49, 201–243 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Dolean, V., Gander, M.J., Gerardo-Giorda, L.: Optimized Schwarz methods for Maxwell’s equations. SIAM J. Sci. Comput. 31(3), 2193–2213 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dolean, V., Gander, M.J., Lanteri, S., Lee, J.-F., Peng, Z.: Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl–curl Maxwell’s equations. J. Comput. Phys. 280, 232–247 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. El Bashir, T.B.A.: Numerical solution of stokes flow generated by vortices: part 2, inside an elliptical cylinder. Acta Mech. 224(11), 2881–2894 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ernst, O.G., Gander, M.J.: Why it is difficult to solve Helmholtz problems with classical iterative methods. Numerical Analysis of Multiscale Problems, pp. 325–363. Springer, Berlin (2012)

    MATH  Google Scholar 

  16. Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gander, M.J.: Schwarz methods over the course of time. Electron. Trans. Numer. Anal. 31(5), 228–255 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Gander, M.J.: On the influence of geometry on optimized Schwarz methods. SeMA J. 53(1), 71–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gander, M.J.: Iterative methods for Helmholtz and Maxwell equations. Math. Forschun. Oberwolfach Rep. 55, 10–11 (2012)

    Google Scholar 

  20. Gander, M.J., Halpern, L.: Absorbing boundary conditions for the wave equation and parallel computing. Math. Comput. 74(249), 153–176 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gander, M.J., Halpern, L.: Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal. 45(2), 666–697 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gander, M.J., Halpern, L., Magoulès, F.: An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation. Int. J. Numer. Methods Fluids 55(2), 163–175 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gander, M.J., Halpern, L., Nataf, F.: Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41(5), 1643–1681 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gander, M.J., Xu, Y.: Optimized Schwarz methods for circular domain decompositions with overlap. SIAM J. Numer. Anal. 52(4), 1981–2004 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gander, M.J., Xu, Y.: Optimized Schwarz method with two-sided transmission conditions in an unsymmetric domain decomposition. In: Domain Decomposition Methods in Science and Engineering XXII, pp. 631–639. (2016)

  26. Gander, M.J., Xu, Y.: Optimized Schwarz methods for model problems with continuously variable coefficients. SIAM J. Sci. Comput. 38(5), A2964–A2986 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gander, M.J., Xu, Y.: Optimized Schwarz methods for domain decompositions with parabolic interfaces. In: Domain Decomposition Methods in Science and Engineering XXIII, pp. 323–331 (2017)

  28. Gander, M.J., Xu, Y.: Optimized Schwarz methods with nonoverlapping circular domain decomposition. Math. Comput. 86(304), 637–660 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gander, M.J., Zhang, H.: Optimized Schwarz methods with overlap for the Helmholtz equation. SIAM J. Sci. Comput. 38(5), A3195–A3219 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gander, M.J., Zhang, H.: A class of iterative solvers for the Helmholtz equation: Factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods. SIAM Rev (2019) (in print)

  31. Gutiérrez-Vega, J.C.: Formal analysis of the propagation of invariant optical fields in elliptic coordinates. PhD thesis, INAOE (2000)

  32. Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  33. von Helmholtz, H.: Theorie der Luftschwingungen in Röhren mit offenen Enden. J. Reine Angew. Math. 57, 1–72 (1860)

    MathSciNet  Google Scholar 

  34. Japhet, C., Nataf, F., Rogier, F.: The optimized order 2 method: application to convection-diffusion problems. Future Generat. Comput. Syst. 18(1), 17–30 (2001)

    Article  MATH  Google Scholar 

  35. Jiang, K., Han, X., Ren, K.: Scattering of a gaussian beam by an elliptical cylinder using the vectorial complex ray model. J. Opt. Soc. Am. A 30(8), 1548–1556 (2013)

    Article  Google Scholar 

  36. Lai, M.: Fast direct solver for poisson equation in a 2d elliptical domain. Numer. Methods Partial Differ. Equ. 20(1), 72–81 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Malyshev, A.N.: Computation of smallest eigenvalues in the Sturm–Liouville problem with strongly varying coefficients. SIAM J. Matrix Anal. Appl. 28(4), 961–970 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Martin, V.: Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations. SIAM J. Sci. Comput. 31(5), 3595–3625 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mitri, F.G.: Acoustic backscattering and radiation force on a rigid elliptical cylinder in plane progressive waves. Ultrasonics 66, 27–33 (2016)

    Article  MathSciNet  Google Scholar 

  40. Patel, S.A., Chhabra, R.P.: Steady flow of Bingham plastic fluids past an elliptical cylinder. J. Non-Newtonian Fluid Mech. 202, 32–53 (2013)

    Article  Google Scholar 

  41. Patel, S.A., Chhabra, R.P.: Laminar free convection in Bingham plastic fluids from an isothermal elliptic cylinder. J. Thermophys. Heat Transf. 30(1), 152–167 (2016)

    Article  Google Scholar 

  42. Patel, S.A., Chhabra, R.P.: Heat transfer in bingham plastic fluids from a heated elliptical cylinder. Int. J. Heat Mass Transf. 73, 671–692 (2014)

    Article  Google Scholar 

  43. Qaddouri, A., Laayouni, L., Loisel, S., Côté, J., Gander, M.J.: Optimized Schwarz methods with an overset grid for the shallow-water equations: preliminary results. Appl. Numer. Math. 58(4), 459–471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rodriguez, A.A., Gerardo-Giorda, L.: New nonoverlapping domain decomposition methods for the harmonic Maxwell system. SIAM J. Sci. Comput. 28(1), 102–122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sheikholeslami, M., Ellahi, R., Hassan, M., Soleimani, S.: A study of natural convection heat transfer in a nanofluid filled enclosure with elliptic inner cylinder. Int. J. Numer. Methods Heat Fluid Flow 24(8), 1906–1927 (2014)

    Article  Google Scholar 

  46. Shen, J., Wang, L.: On spectral approximations in elliptical geometries using Mathieu functions. Math. Comput. 78, 815–844 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Vion, A., Geuzaine, C.: Double sweep preconditioner for optimized schwarz methods applied to the helmholtz problem. J. Comput. Phys. 266, 171–190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, Y.: The influence of domain truncation on the performance of optimized Schwarz methods. Electron. Trans. Numer. Anal. 49, 182–209 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zauderer, E.: Partial Differential Equations of Applied Mathematics, vol. 71. Wiley, New York (2011)

    MATH  Google Scholar 

Download references

Funding

This work is supported by NSFC-12071069, 11671074 and the Fundamental Research Funds for the Central Universities (No. 2412020XK001). This work is supported by NSFC-12071069.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Xin Chen & Yingxiang Xu

  2. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, CP 64, CH-1211, Geneva, Switzerland

    Martin J. Gander

Authors
  1. Xin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin J. Gander
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yingxiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yingxiang Xu.

Ethics declarations

Conflict of interest

No conflicts of interest.

Code Availability

Codes can be provided per requirement.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Gander, M.J. & Xu, Y. Optimized Schwarz Methods with Elliptical Domain Decompositions. J Sci Comput 86, 27 (2021). https://doi.org/10.1007/s10915-020-01394-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01394-8

Keywords

Search

Navigation