High-order finite-difference methods for Poisson’s equation

Author:
H. J. van Linde

Journal:
Math. Comp. **28** (1974), 369-391

MSC:
Primary 65N05

DOI:
https://doi.org/10.1090/S0025-5718-1974-0362936-2

MathSciNet review:
0362936

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- 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 https://doi.org/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** - 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 https://doi.org/10.1145/321250.321260
H. van Linde, - 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** - 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 https://doi.org/10.1007/bf02008824
M. Rockoff, "On the numerical solution of finite difference approximations which are not of positive type,"

*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.

*Notices Amer. Math. Soc.*, v. 10, 1963, p. 108. Abstract #597-169.

Retrieve articles in *Mathematics of Computation*
with MSC:
65N05

Retrieve articles in all journals with MSC: 65N05

Additional Information

Article copyright:
© Copyright 1974
American Mathematical Society