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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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High-order finite-difference methods for Poisson’s equation
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by H. J. van Linde PDF
Math. Comp. 28 (1974), 369-391 Request permission

Abstract:

Finite-difference approximations to the three boundary value problems for Poisson’s equation are given with discretization errors of $O({h^3})$ for the mixed boundary value problem, $O({h^3}|\ln h|)$ for the Neumann problem and $O({h^4})$ for the Dirichlet problem, respectively. These error bounds are an improvement upon similar results obtained by Bramble and Hubbard; moreover, all resulting coefficient matrices are of positive type.
References
  • 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 finite difference analogue of the Neumann problem for Poisson’s equation, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965), 1–14. MR 191107, DOI 10.1137/0702001
  • J. H. Bramble and B. E. Hubbard, Approximation of solutions of mixed boundary value problems for Poisson’s equation by finite differences, J. Assoc. Comput. Mach. 12 (1965), 114–123. MR 171384, DOI 10.1145/321250.321260
  • H. van Linde, High-Order Finite Difference Methods for Poisson’s Equation, Thesis, Groningen, 1971. G. H. Shortley & R. Weller, "The numerical solution of Laplace’s equation," J. Appl. Phys., v. 9, 1938, pp. 334-348.
  • J. H. Bramble and 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. 43 (1964), 117–132. MR 162367, DOI 10.1002/sapm1964431117
  • Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
  • Eduard Batschelet, Über die numerische Auflösung von Ranswertproblemen bei elliptischen partiellen Differentialgleichungen, Z. Angew. Math. Phys. 3 (1952), 165–193 (German). MR 60912, DOI 10.1007/bf02008824
  • M. Rockoff, "On the numerical solution of finite difference approximations which are not of positive type," Notices Amer. Math. Soc., v. 10, 1963, p. 108. Abstract #597-169.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Math. Comp. 28 (1974), 369-391
  • MSC: Primary 65N05
  • DOI: https://doi.org/10.1090/S0025-5718-1974-0362936-2
  • MathSciNet review: 0362936