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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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.
  • 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:
  • MathSciNet review: 0362936