Highly accurate numerical solution of casilinear elliptic boundary-value problems in dimensions

Author:
Victor Pereyra

Journal:
Math. Comp. **24** (1970), 771-783

MSC:
Primary 65.65

DOI:
https://doi.org/10.1090/S0025-5718-1970-0288970-5

MathSciNet review:
0288970

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Abstract | References | Similar Articles | Additional Information

Abstract: The method of Iterated Deferred Corrections, whose theory was developed by the author, is applied to the problems of the title. The necessary asymptotic expansions are obtained and the way in which the corrections are produced by means of numerical differentiation is described in detail. Numerical results and comparisons with the variationalsplines methods are given.

**[1]**Å. Bjőrck & V. Pereyra,*Solution of Vandermonde Systems of Equations*, Pub. 70-02, Dept. de Comp., Fac. Ciencias, Univ. Central de Venezuela, Caracas, 1970;*Math. Comp.*, v. 24, 1970, pp. 893-903. MR**0288970 (44:6165)****[2]**P. G. Ciarlet, M. H. Schultz & R. S. Varga, "Numerical methods of high-order accuracy for nonlinear boundary value problems. V: Monotone operator theory,"*Numer. Math.*, v. 13, 1969, pp. 51-77. MR**0250496 (40:3730)****[3]**J. Daniel, V. Pereyra & L. Schumaker, "Iterated deferred corrections for initial value problems,"*Acta Ci. Venezolana*, v. 19, 1968, pp. 128-135. MR**0255063 (40:8270)****[4]**G. Galimberti & V. Pereyra, "Numerical differentiation and the solution of multidimensional Vandermonde systems," Pub. 69-07, Dept. de Comp., Fac. Ciencias, Univ. Central de Venezuela, Caracas, 1969;*Math. Comp.*, v. 24, 1970, pp. 357-364. MR**0275668 (43:1421)****[5]**R. J. Herbold,*Consistent Quadrature Schemes for the Numerical Solution of Boundary Value Problems by Variational Techniques*, Ph.D. Thesis, Case Western Reserve University, Cleveland, Ohio, 1968.**[6]**V. Pereyra, "Iterated deferred corrections for nonlinear operator equations,"*Numer. Math.*, v. 10, 1967, pp. 316-323. MR**36**#4812. MR**0221760 (36:4812)****[7]**V. Pereyra, "Iterated deferred corrections for nonlinear boundary value problems,"*Numer. Math.*, v. 11, 1968, pp. 111-125. MR**37**#1091. MR**0225498 (37:1091)****[8]**V. Pereyra, "Accelerating the convergence of discretization algorithms,"*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 508-533. MR**36**#4778. MR**0221726 (36:4778)****[9]**V. Pereyra, "On improving an approximate solution of a functional equation by deferred corrections,"*Numer. Math.*, v. 8, 1966, pp. 376-391. MR**34**#3814. MR**0203967 (34:3814)****[10]**C. Pucci,*Some Topics in Parabolic and Elliptic Equations*, Inst. for Fluid Dynamics, Lecture Series, 36, University of Maryland, College Park, Md., 1958.**[11]**L. Bers, "On mildly nonlinear partial differential equations of elliptic type,"*J. Res. Nat. Bur. Standards*, v. 51, 1953, pp. 229-236. MR**16**, 260. MR**0064291 (16:260d)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0288970-5

Keywords:
Casilinear elliptic equations,
boundary-value problems,
deferred corrections,
finite differences,
high-order discrete methods

Article copyright:
© Copyright 1970
American Mathematical Society