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Difference analogues of quasi-linear elliptic Dirichlet problems with mixed derivatives

Author: Robert S. Stepleman
Journal: Math. Comp. 25 (1971), 257-269
MSC: Primary 65N10
MathSciNet review: 0303756
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Abstract: In this paper we consider a class of difference approximations to the Dirichlet problem for second-order quasi-linear elliptic operators with mixed derivative terms. The main result is that for this class of discretizations and bounded g (the right-hand side) a solution to the difference equations exists. We also explicitly exhibit a discretization of this type for a class of operators.

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  • Lipman Bers, On mildly nonlinear partial difference equations of elliptic type, J. Research Nat. Bur. Standards 51 (1953), 229–236. MR 0064291
  • J. H. Bramble and B. E. Hubbard, A theorem on error estimation for finite difference analogues of the Dirichlet problem for elliptic equations, Contributions to Differential Equations 2 (1963), 319–340. MR 152134
  • Lothar Collatz, Numerische Behandlung von Differentialgleichungen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Band LX, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1951 (German). MR 0043563
  • T. Frank, Error Bounds on Numerical Solutions of Dirichlet Problems for Quasi-Linear Elliptic Equations, Thesis, University of Texas, Austin, Tex., 1967.
  • G. T. McAllister, Some nonlinear elliptic partial differential equations and difference equations, J. Soc. Indust. Appl. Math. 12 (1964), 772–777. MR 179958
  • G. T. McAllister, Quasilinear uniformly elliptic partial differential equations and difference equations, SIAM J. Numer. Anal. 3 (1966), no. 1, 13–33. MR 202342, DOI
  • T. S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Physics 31 (1953), 253–259. MR 0052895
  • R. Stepleman, Finite Dimensional Analogues of Variational and Quasi-Linear Elliptic Dirichlet Problems, Thesis, Technical Report #69-88, Computer Science Center, University of Maryland, College Park, Md., 1969.
  • Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502

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Keywords: Difference analogues, elliptic, mixed derivatives
Article copyright: © Copyright 1971 American Mathematical Society