Journal of Mathematical Sciences

, Volume 201, Issue 6, pp 705–732 | Cite as

On a Problem of the Constructive Theory of Harmonic Mappings

  • S. I. Bezrodnykh
  • V. I. Vlasov


The problem of irremovable error appears in finite difference realization of the Winslow approach in the constructive theory of harmonic mappings. As an example, we consider the well-known Roache–Steinberg problem and demonstrate a new approach, which allows us to construct harmonic mappings of complicated domains effectively and with high precision. This possibility is given by the analytic-numerical method of multipoles with exponential convergence rate. It guarantees effective construction of a harmonic mapping with precision controlled by an a posteriori estimate in a uniform norm with respect to the domain.


Dirichlet Problem Conformal Mapping Constructive Theory Grid Generation Harmonic Extension 
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.


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  1. 1.
    L. V. Ahlfors, “Zur Theorie der Überlagerungsflächen,” Acta Math., 65, 157–194 (1935).CrossRefMathSciNetGoogle Scholar
  2. 2.
    L. V. Ahlfors, Lectures on Quasiconformal Mappings, D. Van Nostrand Co., Toronto–New York–London (1966).zbMATHGoogle Scholar
  3. 3.
    G. Alessandrini and V. Nesi, “Invertible harmonic mappings, beyond Kneser,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8, No. 3, 451–468 (2009).Google Scholar
  4. 4.
    G. D. Anderson, M.K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Mappings, Chichester, Wiley (1997).Google Scholar
  5. 5.
    A. S. Arcilla et al. (ed.) “Numerical grid generation in computational fluid dynamics and related fields,” Proceedings, Third International Conference, Barcelona, Spain, 3–7 June, 1991, North-Holland, New York (1991).Google Scholar
  6. 6.
    B.N. Azarenok, “Generation of structured difference grids in two-dimensional nonconvex domains using mappings,” Zh. Vychisl. Mat. Mat. Fiz., 49, No. 5, 797–809 (2009).MathSciNetGoogle Scholar
  7. 7.
    N. S. Bakhvalov, N.P. Zhidkov, and G.M. Kobelkov. Numerical Methods [in Russian], Nauka, Moscow (1987).Google Scholar
  8. 8.
    G. Bateman and A. Erdelyi, Higher Transcendental Functions. Elliptic and Automorphous Functions. Lamé and Mathieu Functions [in Russian], Nauka, Moscow (1967).Google Scholar
  9. 9.
    G. Bateman and A. Erdelyi, Higher Transcendental Functions. Hypergeometric Function. Legendre Functions [in Russian], Nauka, Moscow (1973).Google Scholar
  10. 10.
    P.P. Belinskii, General Properties of Quasiconformal Mappings [in Russian], Nauka, Novosibirsk (1974).Google Scholar
  11. 11.
    P.P. Belinskii, S. K. Godunov, Yu.B. Ivanov, and I.K. Yanenko, “Application of a class of quasiconformal mappings for generation of computational grids in domains with curvilinear boundaries,” Zh. Vychisl. Mat. Mat. Fiz., 15, No. 6, 1499–1511 (1975).Google Scholar
  12. 12.
    L. Bers, “Isolated singularities of minimal surfaces,” Ann. of Math., 53, 364–386 (1951).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    L. Bers, “Univalent solutions of linear elliptic systems,” Comm. Pure Appl. Math., 6, 513–526 (1953).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    S. I. Bezrodnykh, “The singular Riemann–Hilbert problem and its application” [in Russian], PhD thesis, Computational Center Russ. Acad. Sci., Moscow (2006).Google Scholar
  15. 15.
    S. I. Bezrodnykh and V. I. Vlasov, “The singular Riemann–Hilbert problem in a complicated domain,” Spectr. Evol. Probl., 16, 51–62 (2006).Google Scholar
  16. 16.
    B. V. Bojarski, “Homeomorphic solutions of a Beltrami system,” Dokl. Akad. Nauk SSSR, 102, 661–664 (1955).MathSciNetGoogle Scholar
  17. 17.
    B. V. Bojarski and T. Iwaniec, “Quasiconformal mappings and nonlinear elliptic equations in two variables I, II,” Bull. Pol. Acad. Sci. Math., 22, 473–478, 479–484 (1974).zbMATHMathSciNetGoogle Scholar
  18. 18.
    J. U. Brackbill, “An adaptive grid with directional control,” J. Comput. Phys., 108, No. 1, 38–50 (1993).CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    J. U. Brackbill, D. B. Kothe, and H. L. Ruppel, “FLIP: a low-dissipation, particle-in-cell method for fluid flow,” Comput. Phys. Comm., 48, No. 1, 25–38 (1988).CrossRefGoogle Scholar
  20. 20.
    J. U. Brackbill and J. S. Saltzman, “Adaptive zoning for singular problems in two dimensions,” J. Comput. Phys., 46, No. 3, 342–368 (1982).CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    D. Bshouty and W. Hengartner, “Boundary values versus dilatations of harmonic mappings,” J. Anal. Math., 72, 141–164 (1997).CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    D. Bshouty andW. Hengartner, “Univalent harmonic mappings in the plane,” Handbook of Complex Analysis: Geometric Function Theory, Vol. 2, pp. 479–506, Elsevier, Amsterdam (2005).Google Scholar
  23. 23.
    C. Caratheodory, “Über die gegenseitige Beziehung der Ränder bei der Konformer Abbildung des Inneren einer Jordanschen Kurve auf einer Kreis,” Math. Ann., 73, 305–320 (1913).CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    G. Choquet, “Sur un type de transformation analitiques généralisant la représentation conforme et définie au moyen de fonctions harmoniques,” Bull. Cl. Sci. Math. Nat. Sci. Math., 69, No. 2, 156–165 (1945).zbMATHMathSciNetGoogle Scholar
  25. 25.
    W. H. Chu, “Development of a general finite difference approximation for a general domain. I. Mashine transformation,” J. Comput. Phys., 8, 392–408 (1971).CrossRefzbMATHGoogle Scholar
  26. 26.
    J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Ann. Acad. Sci. Fenn. Math., 9, 3–25 (1984).CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    P. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156, Cambridge University Press, Cambridge (2004).Google Scholar
  28. 28.
    P. Duren and D. Khavinson, “Boundary correspondence and dilatation of harmonic mappings,” Complex Variables Theory Appl., 33, 105–111 (1997).CrossRefMathSciNetGoogle Scholar
  29. 29.
    J. Eells and L. Lemaire, “A report on harmonic maps,” Bull. Lond. Math. Soc., 10, 1–68 (1978).CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    P. R. Eiseman, “Adaptive grid generation,” Comput. Methods Appl. Mech. Energ., 64, 321–376 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    S. K. Godunov, A.V. Zabrodin, M.Ya. Ivanov, G. P. Prokopov, and A.M. Krayko, Numerical Solution of Multidimensional Problems of Gas Dynamics [in Russian], Nauka, Moscow (1976).Google Scholar
  32. 32.
    S. K. Godunov and G. P. Prokopov, “The utilization of movable grids in gas dynamic calculations,” Zh. Vychisl. Mat. Mat. Fiz., 12, No. 2, 429–440 (1972).zbMATHMathSciNetGoogle Scholar
  33. 33.
    G. M. Goluzin, Geometrical Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1966).Google Scholar
  34. 34.
    H. Grötzsch, “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes,” Ber. Verh. Sächs. Akad. Wiss., 80, 503–507 (1928).Google Scholar
  35. 35.
    R. R. Hall, “A class of isoperimetric inequalities,” J. Anal. Math., 45, 169–180 (1985).CrossRefzbMATHGoogle Scholar
  36. 36.
    R. Hamilton, Harmonic Maps of Manifolds with Boundary, Lecture Notes in Computer Science, Vol. 471, Springer, Berlin–Heidelberg–New York (1975).Google Scholar
  37. 37.
    E. Heinz, “Über die Lösungen der Minimalflächengleichung,” Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 51–56 (1952).Google Scholar
  38. 38.
    W. Hengartner and G. Schober, “Harmonic mappings with given dilatation,” J. Lond. Math. Soc., 33, 473–483 (1986).CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    W. Hengartner and G. Schober, “On the boundary behavior of orientation-preserving harmonic mappings,” Complex Variables Theory Appl., 5, 197–208 (1986).CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    W. Hengartner and J. Szynal, “Univalent harmonic ring mappings vanishing on the interior boundary,” Can. J. Math., 44, No. 1, 308–323 (1992).CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    J. Hersch and A. Pfluger, “Généralisation du lemme de Schwarz et du principe de la mesure harmonique pour les fonctions pseudo-analytiques,” C. R. Acad. Sci. Paris, 234, 43–45 (1952).zbMATHMathSciNetGoogle Scholar
  42. 42.
    S. A. Ivanenko, “Application of adaptive-harmonic grids for the numerical solution of problems with boundary and interior layers,” Zh. Vychisl. Mat. Mat. Fiz., 35, No. 10, 1494–1517 (1995).MathSciNetGoogle Scholar
  43. 43.
    S. A. Ivanenko, Adaptive Harmonic Grids [in Russian], Computational Center Russ. Acad. Sci., Moscow (1997).Google Scholar
  44. 44.
    S. A. Ivanenko, “Control of cells shape in the course of grid generation,” Zh. Vychisl. Mat. Mat. Fiz., 40, No. 11, 1662–1684 (2000).MathSciNetGoogle Scholar
  45. 45.
    S. A. Ivanenko and A.A. Charakhch’yan, “Curvilinear grids of convex quadrilaterals,” Zh. Vychisl. Mat. Mat. Fiz., 28, No. 4, 503–514 (1988).zbMATHMathSciNetGoogle Scholar
  46. 46.
    J. Jost, Lectures on Harmonic Maps, Lecture Notes in Math., Vol. 1161, Springer, Berlin–New York (1985).Google Scholar
  47. 47.
    M. V. Keldysh and M. A. Lavrentieff, “Sur la représentation conforme des domaines limités par les courbes rectifiables,” Ann. Ecole Norm. Sup. (3), 54, 1–38 (1937).Google Scholar
  48. 48.
    H. Kneser, “Lösung der Aufgabe 41,” J. Ber. Dtsch. Math. Verein., 35, 123–124 (1926).Google Scholar
  49. 49.
    P. Knupp and R. Luczak, “Truncation error in grid generation: a case study,” Numer. Methods Part. Differ. Equ., 11, 561–571 (1995).CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    P. Knupp and S. Steinberg, Fundamentals of Grid Generation, CRC Press, Boca Raton (1993).Google Scholar
  51. 51.
    L.D. Kudryavtsev, “On properties of harmonic mappings of planar domains,” Math. Sb., 36 (78), No. 2, 201–208 (1955).Google Scholar
  52. 52.
    M. Lavrentieff, “Sur une méthode géométrique dans la représentation conforme,” Atti Congr. Intern. Mat. Bologna, 1928: Comm. sez., 3, 241–242, Zanichelli, Bologna (1930).Google Scholar
  53. 53.
    M.A. Lavrentiev, “Sur une classe de représentations continues,” Math. Sb., 42, 407–424 (1935).Google Scholar
  54. 54.
    M. A. Lavrentiev, “A general problem of the theory of quasiconformal representation of plane regions,” Math. Sb., 21 (63), No. 2, 285–320 (1947).Google Scholar
  55. 55.
    M.A. Lavrentiev, “The fundamental theorem of the theory of quasiconformal mappings of twodimensional domains,” Izv. Akad. Nauk SSSR, 12, No. 6, 513–554 (1948).MathSciNetGoogle Scholar
  56. 56.
    O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Springer, Berlin–Heidelberg–New York (1973).CrossRefzbMATHGoogle Scholar
  57. 57.
    H. Lewy, “On the nonvanishing of the Jacobian in certain one-to-one mappings,” Bull. Amer. Math. Soc. (N.S.), 42, 689–692 (1936).CrossRefMathSciNetGoogle Scholar
  58. 58.
    G. Liao, “On harmonic maps,” in: Mathematical Aspects of Numerical Grid Generation (ed.: Castillo J.E.), 123–130, SIAM, Philadelphia (1991).Google Scholar
  59. 59.
    V. D. Liseikin, Grid Generation Methods, Springer, New York, (1999).CrossRefzbMATHGoogle Scholar
  60. 60.
    A. M. Markushevich, Theory of Analytic Functions. Vol. 2 [in Russian], Nauka, Moscow (1968).Google Scholar
  61. 61.
    O. Martio, “On harmonic quasiconformal mappings,” Ann. Acad. Sci. Fenn. Math., 425, 3–10 (1968).MathSciNetGoogle Scholar
  62. 62.
    Ch.B. Jr. Morrey, “On the solutions of quasilinear elliptic partial differential equations,” Trans. Am. Math. Soc., 43, No. 1, 126–166 (1938).CrossRefMathSciNetGoogle Scholar
  63. 63.
    J. C. C. Nitsche, “Über eine mit der Minimalflächengleichung zusammenhängende analytische Funktion und den Bernsteinschen Satz,” Arch. Math. (Basel), 7, 417–419 (1956).CrossRefMathSciNetGoogle Scholar
  64. 64.
    J.C.C. Nitsche, “On an estimate for the curvature of minimal surfaces z = z(x, y),J. Math. Mech., 7, 767–769 (1958).zbMATHMathSciNetGoogle Scholar
  65. 65.
    G.P. Prokopov, “Constructing test problems for generation of two-dimensional regular grids,” Vopr. At. Nauki Tekh. Mat. Model. Fiz. Process., 1, 7–12 (1993).Google Scholar
  66. 66.
    G.P. Prokopov, “Methodology of variational approach to generation of quasiorthogonal grids,” Vopr. At. Nauki Tekh. Mat. Model. Fiz. Process., 1, 37–46 (1998).Google Scholar
  67. 67.
    T. Radó, “Aufgabe 41,” J. Ber. Dtsch. Math. Verein., 45, 49 (1926).Google Scholar
  68. 68.
    T. Radó, “Über den analytischen Charakter der Minimalflächen,” Math. Z., 24, 321–327 (1926).CrossRefMathSciNetGoogle Scholar
  69. 69.
    T. Radó, “Zu einem Satze von S. Bernstein über Minimalflächen im Grossen,” Math. Z., 26, 559–565 (1927).CrossRefzbMATHMathSciNetGoogle Scholar
  70. 70.
    H. Renelt, Elliptic Systems and Quasiconformal Mappings, JohnWiley & Sons, New York (1988).zbMATHGoogle Scholar
  71. 71.
    P. J. Roache and S. Steinberg, “A new approach to grid generation using a variational formulation,” Proc. AIAA 7-th CFD conference, Cincinnati, 360–370 (1985).Google Scholar
  72. 72.
    A. A. Samarskii, Theory of Difference Schemes [in Russian], Nauka, Moscow (1977).Google Scholar
  73. 73.
    S. Sengupta et al. (ed.), Numerical Grid Generation in Computational Fluid Mechanics, Pineridge Press Ltd. (1988).Google Scholar
  74. 74.
    T. I. Serezhnikiva, A.F. Sidorov, and O.V. Ushakova, “On one method of construction of optimal curvilinear grids and its applications,” Sov. J. Numer. Anal. Math. Model., 4, No. 2, 137–155 (1989).Google Scholar
  75. 75.
    T. Sheil-Small, “Constants for planar harmonic mappings,” J. Lond. Math. Soc., 42, 237–248 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  76. 76.
    A. F. Sidorov and T. I. Shabashova, “On a method of calculation of optimal difference grids for multidimensional domains,” Chisl. Metody Mekh. Sploshn. Sredy, 12, No. 5, 106–124 (1981).MathSciNetGoogle Scholar
  77. 77.
    P.W. Smith and S. S. Sritharan, “Theory of harmonic grid generation,” Complex Variables, 10, 359–369 (1988).CrossRefzbMATHMathSciNetGoogle Scholar
  78. 78.
    I. D. Sofronov, V.V. Rasskazova, and L.V. Nesterenko, “Irregular grids in methods of calculation of nonstationary problems of gas dynamics,” in: Vopr. Matem. Modelirovaniya, Vychisl. Mat. Inform., Ministry of Atomic Power of Russia, Moscow–Arzamas-16, 131–183 (1984).Google Scholar
  79. 79.
    S. Steinberg and P. Roache, “Variational curve and surface grid generation,” J. Comput. Phys., 100, No. 1, 163–178 (1992).CrossRefzbMATHMathSciNetGoogle Scholar
  80. 80.
    T. Takagi, K. Miki, B. C. J. Chen, and U. Sha, “Numerical generation of boundary-fitted curvilinear coordinate systems for arbitrary curved surfaces,” J. Comput. Phys., 58, 67–79 (1985).CrossRefzbMATHGoogle Scholar
  81. 81.
    O. Teichmüller, “Eine Anwendung quasikonformen Abbildungen auf das Typenproblem,” Dtsch. Math., 2, 321–327 (1937).Google Scholar
  82. 82.
    O. Teichmüller, “Untersuchungen über konforme und quasikonforme Abbildung,” Dtsch. Math., 3, 621–678 (1938).Google Scholar
  83. 83.
    O. Teichmüller, “Extremal quasikonforme Abbildungen und quadratische Differentiale,” Abh. Preuss. Akad. Wiss., Math., 22, 3–197 (1940).Google Scholar
  84. 84.
    J. F. Thompson (ed.), Numerical Grid Generation, North-Holland, New York (1982).zbMATHGoogle Scholar
  85. 85.
    J. F. Thompson, B.K. Soni, and N.P. Weatherill (ed.), Handbook of Grid Generation, CRC Press, Boca Raton (1999).zbMATHGoogle Scholar
  86. 86.
    J. F. Thompson, Z.U.A. Warsi, and C. W. Mastin, Numerical Grid Generation, North-Holland, New York (1985).zbMATHGoogle Scholar
  87. 87.
    P. N. Vabishchevich, “Composite adaptive meshes in problems of mathematical physics,” Zh. Vychisl. Mat. Mat. Fiz., 29, No. 6, 902–914 (1989).zbMATHGoogle Scholar
  88. 88.
    V. I. Vlasov, “On a method of solving some mixed planar problems for the Laplace equation,” Dokl. Akad. Nauk SSSR, 237, No. 5, 1012–1015 (1977).MathSciNetGoogle Scholar
  89. 89.
    V. I. Vlasov, “Hardy-type space of harmonic functions in domains with angles,” Mat. Vesn., 38, No. 4, 609–616 (1986).zbMATHMathSciNetGoogle Scholar
  90. 90.
    V. I. Vlasov, Boundary-Value Problems in Domains with Curved Boundary [in Russian], Computational Center Russ. Acad. Sci., Moscow (1987).Google Scholar
  91. 91.
    V. I. Vlasov, “Multipole method for solving some boundary value problems in complex-shaped domains,” Z. Angew. Math. Mech., 76, Suppl. 1, 279–282 (1996).zbMATHMathSciNetGoogle Scholar
  92. 92.
    Z. U. Warsi, “Numerical grid generation in arbitrary surfaces through a second-order differentialgeometric model,” J. Comput. Phys., 64, 82–96 (1986).CrossRefzbMATHMathSciNetGoogle Scholar
  93. 93.
    Z. U. Warsi and J. F. Thompson, “Application of variational methods in the fixed and adaptive grid generation,” Comput. Math. Appl., 19, No. 8-9, 31–41 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  94. 94.
    Z. U. Warsi and W. N. Tuarn, “Surface mesh generation using elliptic equations,” in: Numerical Grid Generation in Computational Fluid Dynamics, 95–100, Pineridge Press, UK (1986).Google Scholar
  95. 95.
    W. L. Wendland, Elliptic Systems in the Plane, Pitman, London (1979).zbMATHGoogle Scholar
  96. 96.
    A. Winslow, “Numerical solution of the quasilinear Poisson equations in a nonuniform triangle mesh,” J. Comput. Phys., 2, 149–172 (1967).MathSciNetGoogle Scholar
  97. 97.
    N. N. Yanenko, N.T. Danaev, and V. D. Liseikin, “On a variational method of grid generation,” Chisl. Metody Mekh. Sploshn. Sredy, 8, No. 4, 157–163 (1977).Google Scholar
  98. 98.
    V. A. Zorich, “Quasiconformal maps and the asymptotic geometry of manifolds,” Usp. Mat. Nauk, 57, No. 3, 3–28 (2002).CrossRefMathSciNetGoogle Scholar
  99. 99.
    Surface Modelling, Grid Generation, and Related Issues in Computational Fluid Dynamic Solutions, Proc. Workshop, NASA Lewis Research Center, Cleveland, Ohio, May 9–11, 1995.Google Scholar
  100. 100.
    8th International Conference on Numerical Grid Generation in Computational Field Simulations. Proceedings, Marriott Resort, Waikiki Beach, Honolulu, Hawaii, USA, June 2–6, 2002.Google Scholar
  101. 101.
    14th International Meshing Roundtable. Proceedings, San Diego, USA, 2005, Springer (2005).Google Scholar
  102. 102.
    20th International Meshing Roundtable. Proceedings, Paris, France, 2011, Springer (2011).Google Scholar

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Authors and Affiliations

  1. 1.Dorodnitsyn Computing Center of the Russian Academy of SciencesMoscowRussia
  2. 2.Sternberg Astronomical Institute of the Lomonosov Moscow State UniversityMoscowRussia

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