On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation

Bramble, James H.; Hubbard, B. E.

doi:10.1007/bf01386325

Cited by 112 publications

(112 citation statements)

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“…(4.2) is the reformulation of a "discrete Green's function estimate" in Bramble and Hubbard [6] (see also [1]). Q Proposition 4.1 shows that the full order of consistency is not required near the boundary.…”

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confidence: 99%

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Defect Corrections for Multigrid Solutions of the Dirichlet Problem in General Domains

Auzinger¹

1987

Mathematics of Computation

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Abstract. Recently, the technique of defect correction for the refinement of discrete solutions to elliptic boundary value problems has gained new acceptance in connection with the multigrid approach. In the present paper we give an analysis of a specific application, namely to finite-difference analogues of the Dirichlet problem for Helmholtz's equation, emphasizing the case of nonrectangular domains. A quantitative convergence proof is presented for a class of convex polygonal domains.

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“…For the usual "Shortley-Weller"-operator LA, Proposition 4.1 remains valid without modification (cf. [6]). The discrete regularity properties of the Shortley-Weller scheme have been studied by Hackbusch [11] (see also [1]).…”

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Defect Corrections for Multigrid Solutions of the Dirichlet Problem in General Domains

Auzinger¹

1987

Mathematics of Computation

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“…283].) This method has been refined and extensively used by Collatz [11], Laasonen [16], Bramble and Hubbard [3], [4], Bramble [1], [2], Bramble, Hubbard and Thomée [6], and many others. Moreover, from the maximum principle one can derive estimates to differencequotients of the discrete solution (cf.…”

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confidence: 99%

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1973

Math. Comp.

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Abstract. The generalized local maximum principle for a difference operator L» asserts that if Lam(jc) > 0 then Vu cannot attain its positive maximum at the net-point x. Here r is a local net-operator such that Tu = u + 0(/i) for any smooth function u. This principle, with simple forms of V, is proved for some quite general classes of second-order elliptic operators Lh, whose associated global matrices are not necessarily monotone. It is shown that these generalized principles can be used for easy derivation of global a priori estimates to the solutions of elliptic difference equations and to their difference-quotients. Some examples of parabolic difference equations are also treated.

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“…Bramble and Hubbard [3] showed that the operator lh in (1.4) can be chosen in such a manner that (1.5) holds for v = 4. These authors [4] also constructed operators Lu and lh in the case of a general L (N = 2) such that (1.5) holds with v = 2.…”

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confidence: 99%