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

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Abstract: Finite-difference approximations to the three boundary value problems for Poisson's equation are given with discretization errors of for the mixed boundary value problem, for the Neumann problem and 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.

**[1]**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**0149672**, https://doi.org/10.1007/BF01386325**[2]**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**0191107****[3]**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**0171384**, https://doi.org/10.1145/321250.321260**[4]**H. van Linde,*High-Order Finite Difference Methods for Poisson's Equation*, Thesis, Groningen, 1971.**[5]**G. H. Shortley & R. Weller, "The numerical solution of Laplace's equation,"*J. Appl. Phys.*, v. 9, 1938, pp. 334-348.**[6]**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**0162367****[7]**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502****[8]**Eduard Batschelet,*Über die numerische Auflösung von Ranswertproblemen bei elliptischen partiellen Differentialgleichungen*, Z. Angew. Math. Physik**3**(1952), 165–193 (German). MR**0060912****[9]**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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DOI:
https://doi.org/10.1090/S0025-5718-1974-0362936-2

Article copyright:
© Copyright 1974
American Mathematical Society