Convergence estimates for essentially positive type discrete Dirichlet problems
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- by J. H. Bramble, B. E. Hubbard and Vidar Thomée PDF
- Math. Comp. 23 (1969), 695-709 Request permission
Abstract:
In this paper we consider a class of difference approximations to the Dirichlet problem for second-order elliptic operators with smooth coefficients. The main result is that if the order of accuracy of the approximate problem is $\mathcal {V}$, and $F$ (the right-hand side) and $f$ (the boundary values) both belong to ${C^\lambda }$ for $\lambda < \mathcal {V}$, then the rate of convergence is $O({h^\lambda })$.References
- N. S. Bahvalov, Numerical solution of the Dirichlet problem for Laplace’s equation. , Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959 (1959), no. 5, 171–195 (Russian). MR 0115280
- James H. Bramble, On the convergence of difference approximations for second order uniformly elliptic operators, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 201–209. MR 0260200
- J. H. Bramble and B. E. Hubbard, On the formulation of finite difference analogues of the Dirichlet problem for Poisson’s equation, Numer. Math. 4 (1962), 313–327. MR 149672, DOI 10.1007/BF01386325
- 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
- J. H. Bramble, R. B. Kellogg, and V. Thomée, On the rate of convergence of some difference schemes for second order elliptic equations, Nordisk Tidskr. Informationsbehandling (BIT) 8 (1968), 154–173. MR 238497, DOI 10.1007/bf01933418 L. Collatz, “Bemerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen,” Z. Angew. Math. Mech., v. 13, 1933, pp. 56–57. S. Gerschgorin, “Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen,” Z. Angew. Math. Mech., v. 10, 1930, pp. 373–382.
- Pentti Laasonen, On the solution of Poisson’s difference equation, J. Assoc. Comput. Mach. 5 (1958), 370–382. MR 121999, DOI 10.1145/320941.320951
- Carlo Miranda, Equazioni alle derivate parziali di tipo ellittico, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 2, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (Italian). MR 0087853
- Jaak Peetre and Vidar Thomée, On the rate of convergence for discrete initial-value problems, Math. Scand. 21 (1967), 159–176 (1969). MR 255085, DOI 10.7146/math.scand.a-10856
- E. A. Volkov, Obtaining an error estimate for a numerical solution of the Dirichlet problem in terms of known quantites, Ž. Vyčisl. Mat i Mat. Fiz. 6 (1966), no. 4, suppl., 5–17 (Russian). MR 211625
- E. A. Volkov, Effective error estimates for solutions by the method of nets, of boundary value problems on a rectangle and on certain triangles for the Laplace and Poisson equations, Trudy Mat. Inst. Steklov. 74 (1966), 55–85 (Russian). MR 0225505
- Wolfgang Wasow, On the truncation error in the solution of Laplace’s equation by finite differences, J. Research Nat. Bur. Standards 48 (1952), 345–348. MR 0048923
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Math. Comp. 23 (1969), 695-709
- MSC: Primary 65.66
- DOI: https://doi.org/10.1090/S0025-5718-1969-0266444-7
- MathSciNet review: 0266444