On the numerical solution of the equation \(\frac{{\partial ^2 z}}{{\partial x^2 }}\frac{{\partial ^2 z}}{{\partial y^2 }}  \left( {\frac{{\partial ^2 z}}{{\partial x\partial y}}} \right)^2 = f\) and its discretizations, I
 V. I. Oliker,
 L. D. Prussner
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The equation indicated in the title is the simplest representative of the class of nonlinear equations of MongeAmpere type. Equations with such nonlinearities arise in dynamic meteorology, geometric optics, elasticity and differential geometry. In some special cases heuristic procedures for numerical solution are available, but in order for them to be successful a good initial guess is required. For a bounded convex domain, nonnegativef and Dirichlet data we consider a special discretization of the equation based on its geometric interpretation. For the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence. No special information about an initial guess is required, and to initiate the iterates a routine step is made. The method is selfcorrecting and is structurally suitable for a parallel computer. The computer program modules and several examples are presented in two appendices.
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 Title
 On the numerical solution of the equation \(\frac{{\partial ^2 z}}{{\partial x^2 }}\frac{{\partial ^2 z}}{{\partial y^2 }}  \left( {\frac{{\partial ^2 z}}{{\partial x\partial y}}} \right)^2 = f\) and its discretizations, I
 Journal

Numerische Mathematik
Volume 54, Issue 3 , pp 271293
 Cover Date
 19890501
 DOI
 10.1007/BF01396762
 Print ISSN
 0029599X
 Online ISSN
 09453245
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 AMS (MOS): 65N05
 65G10
 CR: G1.8
 Industry Sectors
 Authors

 V. I. Oliker ^{(1)}
 L. D. Prussner ^{(1)}
 Author Affiliations

 1. Department of Mathematics and Computer Science, Emory University, 30322, Atlanta, GA, USA