Skip to main content
Log in

The development of a grid method for the solution of Laplace's equation in finite or infinite regions with piecewise-smooth boundaries

  • Published:
Mathematical notes of the Academy of Sciences of the USSR Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Literature cited

  1. N. S. Bakhvalov, The numerical solution of the Dirichlet problem for Laplace's equation, Vestn. MGU, Ser. Matem., Mekhan., Astron., Fiz. Khimii, No. 5 (1959), pp. 171–195.

    Google Scholar 

  2. E. A. Volkov, A method of improving the accuracy of the grid method, Dokl. AN SSSR,96, No. 4 (1954), pp. 685–688.

    Google Scholar 

  3. E. A. Volkov, The Solution of the Internal Dirichlet Problem for Laplace's Equation by the Grid Method, In the Collection: “Numerical Mathematics” [in Russian], No. 1 (1957), pp. 34–61.

    Google Scholar 

  4. E. A. Volkov, An Investigation of a Method for Improving the Accuracy of the Grid Method of Solving Poisson's Equation, In the Collection: “Numerical Mathematics” [in Russian], No. 1 (1957), pp. 62–80.

    Google Scholar 

  5. E. A. Volkov, A grid method for solving boundary-value problems involving gradients and normal derivatives, Zh. Vychisl. Matem. i Matem. Fiz.,1, No. 4 (1961), pp. 607–621.

    Google Scholar 

  6. E. A. Volkov, The solution of boundary-value problems for Poisson's equation in a rectangle, Dokl. AN SSSR,147, No. 1 (1962), pp. 13–16.

    Google Scholar 

  7. E. A. Volkov, The use of Lagrange interpolation polynomials in the solution of the Dirichlet problem for Poisson's equation by the grid method, Zh. Vychisl. Matem. i Matem. Fiz.,4, No. 3 (1964), pp. 466–472.

    Google Scholar 

  8. E. A. Volkov, Effective estimates of the error of solutions of the Dirichlet problem for Laplace's equation in a polygon obtained by the grid method, Dokl. AN SSSR,155, No. 4 (1964), pp. 735–738.

    Google Scholar 

  9. E. A. Volkov, The solution of the Dirichlet problem by the method of refining higher order differences, Dokl. AN SSSR,164, No. 3 (1965), pp. 479–482.

    Google Scholar 

  10. E. A. Volkov, Differential properties of solutions of boundary-value problems for the Laplace and Poisson equations in a rectangle, Tr. Matem. in-ta AN SSSR,77 (1965), pp. 89–112.

    Google Scholar 

  11. E. A. Volkov, Differential properties of solutions of boundary-value problems for Laplace's equation in polygons, Tr. Matem. in-ta AN SSSR,77 (1965), pp. 113–142.

    Google Scholar 

  12. E. A. Volkov, The solution of the Dirichlet problem by the method of refining higher order differences, I, Diff. Uravneniya,1, No. 7 (1965), pp. 946–960.

    Google Scholar 

  13. E. A. Volkov, The solution of the Dirichlet problem by the method of refining higher order differences, II, Diff. Uravneniya,1, No. 8 (1965), pp. 1070–1084.

    Google Scholar 

  14. E. A. Volkov, The Estimation in Terms of Known Quantities of the Error in the Numerical Solution of the Dirichlet Problem, In the Collection: “Numerical Methods for the solution of Problems of Mathematical Physics” [in Russian], Moscow (1966), pp. 5–17.

  15. E. A. Volkov, The employment of grid methods in the solution of the external Dirichlet problem, Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 3 (1966), pp. 503–511.

    Google Scholar 

  16. E. A. Volkov, Grid methods with finite and infinite regions with piecewise-smooth boundaries, Dokl. AN SSSR,168, No. 5 (1966), pp. 978–981.

    Google Scholar 

  17. E. A. Volkov, Effective estimates of errors in solutions of boundary-value problems for the Laplace and Poisson equations obtained by means of grid methods in rectangles and certain triangles, Tr. Matem. in-ta AN SSSR,74 (1966), pp. 55–86.

    Google Scholar 

  18. E. A. Volkov, The Combined Grid Method [in Russian], International Mathematics Congress, Moscow (1966) (Short Scientific Communications, Section 14, 28–29).

  19. E. A. Volkov, The non-uniform grid method for finite and infinite regions with conical points, Diff. Uravneniya,2, No. 10 (1966), pp. 1376–1391.

    Google Scholar 

  20. V. A. Kondrat'ev, Boundary-value problems for elliptic equations in finite regions, Dokl. AN SSSR,153, No. 1 (1963), pp. 27–29.

    Google Scholar 

  21. S. M. Nikol'skii, Boundary properties of functions defined in regions with angular points, Matem. Sb.,43, No. 1 (1957), pp. 127–144.

    Google Scholar 

  22. V. V. Fufaev, The Dirichlet problem for regions with corners, Dokl. AN SSSR,131, No. 1 (1960), pp. 37–39.

    Google Scholar 

  23. V. V. Fufaev, Conformal mappings of regions with corners and the differential properties of solutions of Poisson's equation in regions with corners, Dokl. AN SSSR,152, No. 4 (1963), pp. 838–840.

    Google Scholar 

  24. G. I. Éskin, General boundary-value problems for equations of principal type in a plane region with angular points, Uspekhi Matem. Nauk,18, No. 3 (1963), pp. 241–242.

    Google Scholar 

  25. W. G. Bickley, S. Michaelson, and M. R. Osborne, On finite-difference methods for the numerical solution of boundary-value problems, Proc. Roy. Soc. Lond., (A),262 (1961), pp. 219–236.

    Google Scholar 

  26. J. H. Bramble and B. E. Hubbard, A finite difference analog of the Neumann problem for Poisson's equation, J. Soc. Indust. and Appl. Math., Ser. B., Numer. Anal.,2, No. 1 (1965), pp. 1–14.

    Google Scholar 

  27. L. Collatz, Bemerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen, Z. Angew. Math. und Mech.,13, No. 1 (1933), pp. 56–57.

    Google Scholar 

  28. J. H. Giese, On the truncation error in a numerical solution of the Neumann problem for a rectangle, J. Math. Phys.,37, No. 2 (1958), pp. 169–177.

    Google Scholar 

  29. D. Greenspan, Introductory numerical analysis of elliptic boundary-value problems, New York-Evanston-London (1965).

  30. P. Laasonen, On the solution of Poisson's difference equation, J. Assoc. Comput. Mach.,5, No. 4, (1958), pp. 370–382.

    Google Scholar 

  31. W. Wasow, On the truncation error in the solution of Laplace's equation by finite differences, J. Res. Nat. Bur. Standards,48 (1952), pp. 345–348.

    Google Scholar 

Download references

Authors

Additional information

Translated from Matematicheskie Zametki, Vol. 2, No. 4, pp. 421–435, October, 1967.

The author takes this opportunity to thank sincerely S. M. Nikol'skii for his constant interest in this work and his valuable advice.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Volkov, E.A. The development of a grid method for the solution of Laplace's equation in finite or infinite regions with piecewise-smooth boundaries. Mathematical Notes of the Academy of Sciences of the USSR 2, 747–755 (1967). https://doi.org/10.1007/BF01093653

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01093653

Keywords

Navigation