Skip to main content
SpringerLink
Log in

The Method of Harmonic Mapping of Regions with a Notch

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

It is well known that common numerical methods encounter serious computational difficulties when constructing harmonic mappings of regions with notches and generating computational grids in such regions on the basis of these mappings. We propose an efficient computational technique for constructing a harmonic mapping of domains with a rectangular notch based on the analytical–numerical multipole method. Our research demonstrates high computational efficiency of the proposed method; the use of only \(40\) approximative functions (multipoles) provides an error of less than \(10^{-7}\) in the \(C (\overline{g})\) norm. The result is obtained with the help of a posteriori estimation. We find a condition ensuring that the grid line issuing from the vertex of an angle of magnitude \(\pi \beta\), \(\beta > 1\), is not tangent to the angle sides at this vertex, which prevents the emergence of an adverse grid self-overlapping effect.

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

Similar content being viewed by others

References

  1. T. Radó, “Aufgabe 41,” Jahresbericht Deutsch Math. Verein 35, 49 (1926).

    Google Scholar 

  2. H. Kneser, “Lösung der Aufgabe 41,” Jahresbericht Deutsch. Math. Verein 35, 123–124 (1926).

    Google Scholar 

  3. G. Choquet, “Sur un type de transformation analitiques généralisant la représentation conforme et définie au moyen de fonctions harmoniques,” Bull. Sci. Math 69 (2), 156–165 (1945).

    MathSciNet  MATH  Google Scholar 

  4. R. Hamilton, “Harmonic maps of manifolds with boundary,” in Lecture Notes in Mathematics (Springer- Verlag, Berlin, Heidelberg–New York, 1975), No. 471.

    Google Scholar 

  5. J. Jost, “Harmonic mappings and minimal immersions,” in Lecture Notes in Mathematics (Springer- Verlag, Berlin–New York, 1985), No. 1161, pp. 118–192.

    Google Scholar 

  6. P. Duren, “Harmonic mappings in the plane,” in Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 2004), No. 156.

    Google Scholar 

  7. A. M. Winslow, “Numerical solution of the quazi–linear Poisson equations in a nonuniform triangle mesh,” J. Comp. Phys. 2 (2), 149–172 (1966–1967).

    Article  MATH  Google Scholar 

  8. S. K. Godunov and G. P. Prokopov, “On the use of mooving grids in gas dynamic computations.,” Comp. Math. Math. Phys. 12 (2), 429-440 (1972).

    MATH  Google Scholar 

  9. J. F. Thompson, F. C. Thames, and C. W. Mastin, “Automatic numerical generation of body fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies,” J. Comp. Phys. 15, 299-319 (1974).

    Article  MATH  Google Scholar 

  10. S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, G. P. Prokopov, and A. M. Krayko, Numerical Solution of Multidimensional Problems of Gas Dynamics (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  11. J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin, Numerical Grid Generation (North-Holland, New York, 1985).

    MATH  Google Scholar 

  12. S. Sengupta et al.(Eds.), Numerical Grid Generation in Computational Fluid Mechanics ’88 (Pineridge Press, Swansea, UK, 1988).

    Google Scholar 

  13. P. Knupp and S. Steinberg, Fundamentals of Grid Generation. (CRC Press, Boca Raton, FL, 1993).

    MATH  Google Scholar 

  14. S. A. Ivanenko, Adaptive Harmonic Meshes (Computing Center of the RAS, Moscow, 1997).

    MATH  Google Scholar 

  15. V. D. Liseikin, Methods for Constructing Difference Grids (Novosibirsk State Univ., Novosibirsk, 2014) [in Russian].

    Google Scholar 

  16. J. U. Brackbill and J. S. Saltzman, “Adaptive zoning for singular problems in two dimensions,” J. Comput. Phys 46 (3), 342–368 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  17. S. A. Ivanenko and A. A. Charakhch’yan, “Curvilinear grids of convex quadrilaterals,” Comp. Math. Math. Phys. 28 (4), 503–514 (1988).

    MATH  Google Scholar 

  18. G. P. Prokopov, “Methodology of variational approach to generation of quasiorthogonal grids,” Vopr. At. Nauki Tekh. Mat. Model. Fiz. Process, No. 1, 37–46 (1998) [in Russian].

    Google Scholar 

  19. J. F. Thompson, B. K. Soni, and N. P. Weatherill (Eds.), Handbook of Grid Generation (CRC Press, Boca Raton, FL, 1999).

    Google Scholar 

  20. J. E. Castillo, S. Steinberg, and P. J. Roache, “Mathematical aspects of variational grid generation II,” J. Comp. and Appl. Math. 20, 127–135 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  21. J. E. Castillo, P. J. Roache, and S. Steinberg, “On the folding of numerically generated grids: use of a reference grid,” Communs Appl. Numer. Meth 4, 471–481 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. A. Ivanenko, “Control of cells shapes in the course of grid generation,” Comp. Math. Math. Phys. 40 (11), 1662–1684 (2000).

    MATH  Google Scholar 

  23. P. J. Roache and S. Steinberg, “A new approach to grid genaration using a variational formulation,” Proc. AIAA 7–th Computational Fluid Dynamics Conference, No. Cincinnati OH, July, 360–370 (1985).

  24. P. Knupp and R. Luczak, “Truncation error in grid generation: a case study,” Numer. Meths Part. Differ. Equats 11, 561–571 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  25. B. N. Azarenok, “Generation of structured difference grids in two–dimensional nonconvex domains using mappings,” Comp. Math. Math. Phys. 49 (5), 826–839 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. I. Bezrodnykh and V. I. Vlasov, “On the behavior of harmonic mappings in angles,” Math. Notes 101 (3), 474–480 (2017).

    MathSciNet  MATH  Google Scholar 

  27. S. I. Bezrodnykh and V. I. Vlasov, “Singular behavior of harmonic maps near corners,” Complex Variables and Elliptic Equtions 64 (5), 838–851 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  28. S. I. Bezrodnykh and V. I. Vlasov, “On a computational problem of two-dimensional harmonic maps,” Scientific Bulletin of BelSU, No. 13 (68), 30–44 (2009) [in Russian].

    Google Scholar 

  29. S. I. Bezrodnykh and V. I. Vlasov, “On a problem of the constructive theory of harmonic mappings,” Journal of Mathematical Sciences 46 (6), 705–732 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  30. V. I. Vlasov, “On a method of solving some mixed planar problems for the Laplace equation,” Dokl. Akad. Nauk SSSR 237 (5), 1012–1015 (1977) [in Russian].

    MathSciNet  Google Scholar 

  31. V. I. Vlasov, Boundary Value Problems in Domains with Curvilinear Boundary, Doctoral thesis (Comput. Center Acad. Sci. SSSR, Moscow, 1990) [in Russian].

    Google Scholar 

  32. V. I. Vlasov, “Multipole method for solving some boundary value problems in complex–shaped domains,” Zeitschr. Angew. Math. Mech. 76 (Suppl. 1), 279–282 (1996).

    MathSciNet  MATH  Google Scholar 

  33. M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of Complex Variables (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  34. G. I. Arkhipov, V. A. Sadovnichy, and V. N. Chubarikov, Lectures in Mathematical Analysis (Moscow University, Moscow, 2004) [in Russian].

    Google Scholar 

  35. N. S. Bakhvalov, N. P. Zhidkov, and G. I. Kobelkov, Numerical Methods (Nauka, Moscow, 1987) [in Russian].

    Google Scholar 

Download references

Funding

The paper was published with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement 075-15-2022-284.

Author information

Authors and Affiliations

  1. Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Moscow, 119333, Russia

    S. I. Bezrodnykh & V. I. Vlasov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    V. I. Vlasov

Authors
  1. S. I. Bezrodnykh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. I. Vlasov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. I. Bezrodnykh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bezrodnykh, S.I., Vlasov, V.I. The Method of Harmonic Mapping of Regions with a Notch. Math Notes 112, 831–844 (2022). https://doi.org/10.1134/S0001434622110189

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434622110189

Keywords

Search

Navigation