Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

The exact order of convergence for finite difference approximations to ordinary boundary value problems


Author: Wolf-Jürgen Beyn
Journal: Math. Comp. 33 (1979), 1213-1228
MSC: Primary 65L10
DOI: https://doi.org/10.1090/S0025-5718-1979-0537966-2
MathSciNet review: 537966
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the problem of determining the exact order of convergence for the finite difference method applied to ordinary boundary value problems when formulas of different orders are used at different points of the grid. Under rather general assumptions, it is shown that the global discretization error is $ O({h^\tau })$ if the local truncation error is $ O({h^\tau })$ on the boundary and at interior grid points, while it is only $ O({h^{\tau - (k - \mu )}})$ at grid points near the boundary. Here k and $ \mu $ denote the order of the differential and the boundary operator, respectively.


References [Enhancements On Off] (What's this?)

  • [1] W.-J. BEYN, "Zur Stabilität von Differenzenverfahren für Systeme linearer gewöhnlicher Randwertaufgaben," Numer. Math., v. 29, 1978, pp. 209-226. MR 0488777 (58:8288)
  • [2] E. BOHL, Zur Anwendung von Differenzenschemen mit symmetrischen Formeln bei Randwertaufgaben, ISNM 32, Birkhäuser Verlag, Berlin, 1976, pp. 25-47. MR 0519865 (58:24970)
  • [3] J. H. BRAMBLE & B. E. HUBBARD, "On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation," Numer. Math., v. 4, 1962, pp. 313-327. MR 0149672 (26:7157)
  • [4] J. H. BRAMBLE & B. E. HUBBARD, "On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type," J. Math. and Phys., v. 43, 1964, pp. 117-132. MR 0162367 (28:5566)
  • [5] J. H. BRAMBLE & B. E. HUBBARD, "New monotone type approximations for elliptic problems," Math. Comp., v. 18, 1964, pp. 349-367. MR 0165702 (29:2982)
  • [6] P. G. CIARLET, "Discrete maximum principle for finite-difference operators," Aequationes Math., v. 4, 1970, pp. 338-352. MR 0292317 (45:1404)
  • [7] H. ESSER, "Stabilitätsungleichungen für Diskretisierungen von Randwertaufgaben gewöhnlicher Differentialgleichungen," Numer. Math., v. 28, 1977, pp. 69-100. MR 0461926 (57:1908)
  • [8] R. D. GRIGORIEFF, "Die Konvergenz des Rand- und Eigenwertproblems linearer gewöhnlicher Differenzengleichungen," Numer. Math., v. 15, 1970, pp. 15-48. MR 0270568 (42:5456)
  • [9] N. N. GUDOVICH, "A new method for constructing stable finite-difference schemes of any previously assigned order of approximation for linear ordinary differential equations," U.S.S.R. Computational Math. and Math. Phys., v. 15 (4), 1975, pp. 115-129.
  • [10] P. HENRICI, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, London, Sydney, 1962. MR 0135729 (24:B1772)
  • [11] L. W. KANTOROWITSCH & G. P. AKILOW, Funktionalanalysis in normierten Räumen, Akademie Verlag, Berlin, 1964. MR 0177273 (31:1536)
  • [12] H.-O. KREISS, "Difference approximations for boundary and eigenvalue problems for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 605-624. MR 0373296 (51:9496)
  • [13] J. LORENZ, Die Inversmonotonie von Matrizen und ihre Anwendung beim Stabilitätsnachweis von Differenzenverfahren, Ph. D. Thesis, University of Münster, 1975.
  • [14] J. LORENZ, "Zur Inversmonotonie diskreter Probleme," Numer. Math., v. 27, 1977, pp. 227-238. MR 0468138 (57:7976)
  • [15] H. S. PRICE, "Monotone and oscillation matrices applied to finite difference approximations," Math. Comp., v. 22, 1968, pp. 489-516. MR 0232550 (38:875)
  • [16] H. J. STETTER, Analysis of Discretization Methods for Ordinary Differential Equations, Springer Tracts in Natural Philosophy, Vol. 23, Springer-Verlag, Berlin, Heidelberg, New York, 1973. MR 0426438 (54:14381)
  • [17] V. THOMÉE & B. WESTERGREN, "Elliptic difference equations and interior regularity," Numer. Math., v. 11, 1968, pp. 196-210. MR 0224303 (36:7347)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L10

Retrieve articles in all journals with MSC: 65L10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1979-0537966-2
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society