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

DOI:
https://doi.org/10.1090/S0025-5718-1971-0303756-1

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.

**[1]**Lipman Bers,*On mildly nonlinear partial difference equations of elliptic type*, J. Research Nat. Bur. Standards**51**(1953), 229–236. MR**0064291****[2]**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**0152134****[3]**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****[4]**T. Frank,*Error Bounds on Numerical Solutions of Dirichlet Problems for Quasi-Linear Elliptic Equations*, Thesis, University of Texas, Austin, Tex., 1967.**[5]**G. T. McAllister,*Some nonlinear elliptic partial differential equations and difference equations*, J. Soc. Indust. Appl. Math.**12**(1964), 772–777. MR**0179958****[6]**G. T. McAllister,*Quasilinear uniformly elliptic partial differential equations and difference equations*, SIAM J. Numer. Anal.**3**(1966), no. 1, 13–33. MR**0202342**, https://doi.org/10.1137/0703002**[7]**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****[8]**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.**[9]**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0303756-1

Keywords:
Difference analogues,
elliptic,
mixed derivatives

Article copyright:
© Copyright 1971
American Mathematical Society