Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. V. Ahlfors, “Zur Theorie der Überlagerungsflächen,” Acta Math., 65, 157–194 (1935).
L. V. Ahlfors, Lectures on Quasiconformal Mappings, D. Van Nostrand Co., Toronto–New York–London (1966).
G. Alessandrini and V. Nesi, “Invertible harmonic mappings, beyond Kneser,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8, No. 3, 451–468 (2009).
G. D. Anderson, M.K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Mappings, Chichester, Wiley (1997).
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).
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).
N. S. Bakhvalov, N.P. Zhidkov, and G.M. Kobelkov. Numerical Methods [in Russian], Nauka, Moscow (1987).
G. Bateman and A. Erdelyi, Higher Transcendental Functions. Elliptic and Automorphous Functions. Lamé and Mathieu Functions [in Russian], Nauka, Moscow (1967).
G. Bateman and A. Erdelyi, Higher Transcendental Functions. Hypergeometric Function. Legendre Functions [in Russian], Nauka, Moscow (1973).
P.P. Belinskii, General Properties of Quasiconformal Mappings [in Russian], Nauka, Novosibirsk (1974).
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).
L. Bers, “Isolated singularities of minimal surfaces,” Ann. of Math., 53, 364–386 (1951).
L. Bers, “Univalent solutions of linear elliptic systems,” Comm. Pure Appl. Math., 6, 513–526 (1953).
S. I. Bezrodnykh, “The singular Riemann–Hilbert problem and its application” [in Russian], PhD thesis, Computational Center Russ. Acad. Sci., Moscow (2006).
S. I. Bezrodnykh and V. I. Vlasov, “The singular Riemann–Hilbert problem in a complicated domain,” Spectr. Evol. Probl., 16, 51–62 (2006).
B. V. Bojarski, “Homeomorphic solutions of a Beltrami system,” Dokl. Akad. Nauk SSSR, 102, 661–664 (1955).
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).
J. U. Brackbill, “An adaptive grid with directional control,” J. Comput. Phys., 108, No. 1, 38–50 (1993).
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).
J. U. Brackbill and J. S. Saltzman, “Adaptive zoning for singular problems in two dimensions,” J. Comput. Phys., 46, No. 3, 342–368 (1982).
D. Bshouty and W. Hengartner, “Boundary values versus dilatations of harmonic mappings,” J. Anal. Math., 72, 141–164 (1997).
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).
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).
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).
W. H. Chu, “Development of a general finite difference approximation for a general domain. I. Mashine transformation,” J. Comput. Phys., 8, 392–408 (1971).
J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Ann. Acad. Sci. Fenn. Math., 9, 3–25 (1984).
P. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156, Cambridge University Press, Cambridge (2004).
P. Duren and D. Khavinson, “Boundary correspondence and dilatation of harmonic mappings,” Complex Variables Theory Appl., 33, 105–111 (1997).
J. Eells and L. Lemaire, “A report on harmonic maps,” Bull. Lond. Math. Soc., 10, 1–68 (1978).
P. R. Eiseman, “Adaptive grid generation,” Comput. Methods Appl. Mech. Energ., 64, 321–376 (1987).
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).
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).
G. M. Goluzin, Geometrical Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1966).
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).
R. R. Hall, “A class of isoperimetric inequalities,” J. Anal. Math., 45, 169–180 (1985).
R. Hamilton, Harmonic Maps of Manifolds with Boundary, Lecture Notes in Computer Science, Vol. 471, Springer, Berlin–Heidelberg–New York (1975).
E. Heinz, “Über die Lösungen der Minimalflächengleichung,” Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 51–56 (1952).
W. Hengartner and G. Schober, “Harmonic mappings with given dilatation,” J. Lond. Math. Soc., 33, 473–483 (1986).
W. Hengartner and G. Schober, “On the boundary behavior of orientation-preserving harmonic mappings,” Complex Variables Theory Appl., 5, 197–208 (1986).
W. Hengartner and J. Szynal, “Univalent harmonic ring mappings vanishing on the interior boundary,” Can. J. Math., 44, No. 1, 308–323 (1992).
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).
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).
S. A. Ivanenko, Adaptive Harmonic Grids [in Russian], Computational Center Russ. Acad. Sci., Moscow (1997).
S. A. Ivanenko, “Control of cells shape in the course of grid generation,” Zh. Vychisl. Mat. Mat. Fiz., 40, No. 11, 1662–1684 (2000).
S. A. Ivanenko and A.A. Charakhch’yan, “Curvilinear grids of convex quadrilaterals,” Zh. Vychisl. Mat. Mat. Fiz., 28, No. 4, 503–514 (1988).
J. Jost, Lectures on Harmonic Maps, Lecture Notes in Math., Vol. 1161, Springer, Berlin–New York (1985).
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).
H. Kneser, “Lösung der Aufgabe 41,” J. Ber. Dtsch. Math. Verein., 35, 123–124 (1926).
P. Knupp and R. Luczak, “Truncation error in grid generation: a case study,” Numer. Methods Part. Differ. Equ., 11, 561–571 (1995).
P. Knupp and S. Steinberg, Fundamentals of Grid Generation, CRC Press, Boca Raton (1993).
L.D. Kudryavtsev, “On properties of harmonic mappings of planar domains,” Math. Sb., 36 (78), No. 2, 201–208 (1955).
M. Lavrentieff, “Sur une méthode géométrique dans la représentation conforme,” Atti Congr. Intern. Mat. Bologna, 1928: Comm. sez., 3, 241–242, Zanichelli, Bologna (1930).
M.A. Lavrentiev, “Sur une classe de représentations continues,” Math. Sb., 42, 407–424 (1935).
M. A. Lavrentiev, “A general problem of the theory of quasiconformal representation of plane regions,” Math. Sb., 21 (63), No. 2, 285–320 (1947).
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).
O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Springer, Berlin–Heidelberg–New York (1973).
H. Lewy, “On the nonvanishing of the Jacobian in certain one-to-one mappings,” Bull. Amer. Math. Soc. (N.S.), 42, 689–692 (1936).
G. Liao, “On harmonic maps,” in: Mathematical Aspects of Numerical Grid Generation (ed.: Castillo J.E.), 123–130, SIAM, Philadelphia (1991).
V. D. Liseikin, Grid Generation Methods, Springer, New York, (1999).
A. M. Markushevich, Theory of Analytic Functions. Vol. 2 [in Russian], Nauka, Moscow (1968).
O. Martio, “On harmonic quasiconformal mappings,” Ann. Acad. Sci. Fenn. Math., 425, 3–10 (1968).
Ch.B. Jr. Morrey, “On the solutions of quasilinear elliptic partial differential equations,” Trans. Am. Math. Soc., 43, No. 1, 126–166 (1938).
J. C. C. Nitsche, “Über eine mit der Minimalflächengleichung zusammenhängende analytische Funktion und den Bernsteinschen Satz,” Arch. Math. (Basel), 7, 417–419 (1956).
J.C.C. Nitsche, “On an estimate for the curvature of minimal surfaces z = z(x, y),” J. Math. Mech., 7, 767–769 (1958).
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).
G.P. Prokopov, “Methodology of variational approach to generation of quasiorthogonal grids,” Vopr. At. Nauki Tekh. Mat. Model. Fiz. Process., 1, 37–46 (1998).
T. Radó, “Aufgabe 41,” J. Ber. Dtsch. Math. Verein., 45, 49 (1926).
T. Radó, “Über den analytischen Charakter der Minimalflächen,” Math. Z., 24, 321–327 (1926).
T. Radó, “Zu einem Satze von S. Bernstein über Minimalflächen im Grossen,” Math. Z., 26, 559–565 (1927).
H. Renelt, Elliptic Systems and Quasiconformal Mappings, JohnWiley & Sons, New York (1988).
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).
A. A. Samarskii, Theory of Difference Schemes [in Russian], Nauka, Moscow (1977).
S. Sengupta et al. (ed.), Numerical Grid Generation in Computational Fluid Mechanics, Pineridge Press Ltd. (1988).
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).
T. Sheil-Small, “Constants for planar harmonic mappings,” J. Lond. Math. Soc., 42, 237–248 (1990).
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).
P.W. Smith and S. S. Sritharan, “Theory of harmonic grid generation,” Complex Variables, 10, 359–369 (1988).
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).
S. Steinberg and P. Roache, “Variational curve and surface grid generation,” J. Comput. Phys., 100, No. 1, 163–178 (1992).
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).
O. Teichmüller, “Eine Anwendung quasikonformen Abbildungen auf das Typenproblem,” Dtsch. Math., 2, 321–327 (1937).
O. Teichmüller, “Untersuchungen über konforme und quasikonforme Abbildung,” Dtsch. Math., 3, 621–678 (1938).
O. Teichmüller, “Extremal quasikonforme Abbildungen und quadratische Differentiale,” Abh. Preuss. Akad. Wiss., Math., 22, 3–197 (1940).
J. F. Thompson (ed.), Numerical Grid Generation, North-Holland, New York (1982).
J. F. Thompson, B.K. Soni, and N.P. Weatherill (ed.), Handbook of Grid Generation, CRC Press, Boca Raton (1999).
J. F. Thompson, Z.U.A. Warsi, and C. W. Mastin, Numerical Grid Generation, North-Holland, New York (1985).
P. N. Vabishchevich, “Composite adaptive meshes in problems of mathematical physics,” Zh. Vychisl. Mat. Mat. Fiz., 29, No. 6, 902–914 (1989).
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).
V. I. Vlasov, “Hardy-type space of harmonic functions in domains with angles,” Mat. Vesn., 38, No. 4, 609–616 (1986).
V. I. Vlasov, Boundary-Value Problems in Domains with Curved Boundary [in Russian], Computational Center Russ. Acad. Sci., Moscow (1987).
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).
Z. U. Warsi, “Numerical grid generation in arbitrary surfaces through a second-order differentialgeometric model,” J. Comput. Phys., 64, 82–96 (1986).
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).
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).
W. L. Wendland, Elliptic Systems in the Plane, Pitman, London (1979).
A. Winslow, “Numerical solution of the quasilinear Poisson equations in a nonuniform triangle mesh,” J. Comput. Phys., 2, 149–172 (1967).
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).
V. A. Zorich, “Quasiconformal maps and the asymptotic geometry of manifolds,” Usp. Mat. Nauk, 57, No. 3, 3–28 (2002).
Surface Modelling, Grid Generation, and Related Issues in Computational Fluid Dynamic Solutions, Proc. Workshop, NASA Lewis Research Center, Cleveland, Ohio, May 9–11, 1995.
8th International Conference on Numerical Grid Generation in Computational Field Simulations. Proceedings, Marriott Resort, Waikiki Beach, Honolulu, Hawaii, USA, June 2–6, 2002.
14th International Meshing Roundtable. Proceedings, San Diego, USA, 2005, Springer (2005).
20th International Meshing Roundtable. Proceedings, Paris, France, 2011, Springer (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 46, Proceedings of the Sixth International Conference on Differential and Functional Differential Equations and International Workshop “Spatio-Temporal Dynamical Systems” (Moscow, Russia, 14–21 August, 2011). Part 2, 2012.
Rights and permissions
About this article
Cite this article
Bezrodnykh, S.I., Vlasov, V.I. On a Problem of the Constructive Theory of Harmonic Mappings. J Math Sci 201, 705–732 (2014). https://doi.org/10.1007/s10958-014-2021-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-2021-x