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On the numerical solution of the equation \(\frac{\partial ^ 2z\partial ^ 2z}{\partial x^ 2\partial y^ 2}-(\frac{\partial ^ 2z}{\partial x\partial y})^ 2=f\) and its discretizations. I. (English) Zbl 0659.65116

For a bounded convex domain, nonnegative f and Dirichlet data a special discretization of the indicated of the title equation is constructed. For the discrete version of the problem an iterative method that produces a monotonically convergent sequences is proposed. Several numerical examples are presented.
Reviewer: L.G.Vulkov

MSC:

65Z05 Applications to the sciences
35Q99 Partial differential equations of mathematical physics and other areas of application
65N06 Finite difference methods for boundary value problems involving PDEs
65G30 Interval and finite arithmetic
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References:

[1] Aleksandrov, A.D.: The Dirichlet problem for the equation det(z ij )=?(z 1,...,z n,z, x 1,...,x n). I. Vestnik Leningrad Univ.1 5-24 (1958) (in Russian) · Zbl 0114.30202
[2] Aleksandrov, A.D.: Convex polyhedra. GITTL, M.-L. 1950 (in Russian); German transl. Berlin, Akademie-Verlag 1958
[3] Arnason, G.: A convergent method for solving the balance equation. J. Meteorol.15, 220-225 (1957)
[4] Bakelman, I.: Geometric methods for solving elliptic equations. Nauka 1965 (in Russian)
[5] Busemann, H.: Convex surfaces. New York, Interscience Publishers, 1958 · Zbl 0196.55101
[6] Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math.37, 369-402 (1984) · Zbl 0598.35047
[7] Cheng, S.Y., Yau, S.T.: On the regularity of the Monge-Ampère equation det(?2 u/? x i?y j )=F(x,u). Comm. Pure Appl. Math.30 41-68 (1977) · Zbl 0347.35019
[8] Courant, R., Hilbert, D.: Methods of mathematical physics. Vol. II, New York: Interscience Publishers, Wiley 1962 · Zbl 0099.29504
[9] Haltiner, G.J.: Numerical weather prediction, New York, Wiley 1971
[10] Kasahara, A.: Significance of non-elliptic regions in balanced flows of the tropical atmosphere (preprint). Boulder, Colorado, National Center for Atmospheric Research, 1981
[11] Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math.6, 337-394 (1953) · Zbl 0051.12402
[12] Oliker, V.: On the linearized Monge-Ampère equations related to the boundary value Minkowski problem and its generalizations. In: Gherardelli, F. (ed.) Monge-Ampère Equations and Related Topics, Proceedings of a Seminar held in Firenze, 1980, pp. 79-112 Roma 1982
[13] Ortega, J.M., Rheinboldt, W.C.: Iterative solutions of nonlinear equations in several variables. New York: Academic Press 1970 · Zbl 0241.65046
[14] Pogorelov, A.V.: Deformation of convex surfaces. GITTL. M.-L. 1951 (in Russian) (see esp. Ch. II, Sect. 4)
[15] Pogorelov, A.V.: The Minkowski multidimensional problem. Moscow, Nauka 1975 (in Russian); Engl. transl. New York: Wiley J. 1978
[16] Rheinboldt, W.: On M-functions and their application to nonlinear Gauss-Seidel iterations and to network flows. J. Math. Anal. Appl.32, 274-307 (1970) · Zbl 0206.46504
[17] Rheinboldt, W.: Methods for solving systems of nonlinear equations. Regional Conference Series in Applied Math. # 14. SIAM. Philadelphia 1974 · Zbl 0325.65022
[18] Shuman, F.G.: Numerical methods in weather prediction: I. The Balance equation. Monthly Weather Review85, 329-332 (1957)
[19] Stoker, J.J.: Nonlinear elasticity. Ch. 5.. New York: Gordon and Breach 1968 · Zbl 0187.45801
[20] Swart, G.: Finding the convex hull facet by facet. Algorithms6, 17-48 (1985) · Zbl 0563.68041
[21] Volkov, Y.A.: An estimate for the change of solution of the equationf(z 1, ...,z n ) det(z ij )=h(x 1, ...,x n ) in terms of the change of its right hand side. Vestnik Leningrad Univ.13 5-14 (1960) (in Russian)
[22] Westcott, B.S.: Shaped reflector antenna design. Research Studies Press Ltd., Letchworth, Hertforedshire, England 1983
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