On the convergence of the conjugate gradient method for singular capacitance matrix equations from the Neumann problem of the Poisson equation

Abstract

Summary.It is shown analytically in this work that the conjugate gradient method is an efficient means of solving the singular capacitance matrix equations arising from the Neumann problem of the Poisson equation. The total operation count of the algorithm does not exceed constant times N21ogN(N=l/h) for any bounded domain with sufficiently smooth boundary.

Untitled

“…See Widlund [57] and Proskurowski and Widlund [44] for further discussion. Such arguments are also central in the work of Shieh [47]. He was able to prove that all except a fixed number of singular values of certain capacitance matrices for problems in the plane lie in a fixed interval while the remaining few are no closer than Khq, K and q constants, from the origin.…”

“…The usefulness of these algorithms has been extended in recent years to problems on general bounded regions by the development of capacitance matrix, or imbedding, methods; see Buzbee and Dorr [6], Buzbee, Dorr, George and Golub [7] [42], [43], Proskurowski and Widlund [44], [45], Shieh [46], [47], [48] and Widlund [57]. We refer to Proskurowski and Widlund [44] for a discussion of this development up to the beginning of 1976.…”

“…All of the numerical experiments reported in those papers were carried out for regions in the plane. Strong results on the efficiency of certain of these methods have been rigorously established through the excellent work of Shieh [46], [47], [48]. Algorithms similar to those which we shall describe have recently been implemented very successfully for two-dimensional regions by Proskurowski [42], [43] and Proskurowski and Widlund [45].…”

On finite element methods for the Neumann problem

“…[10,12,14,16,17,22]). The corresponding methods for the Neumann problem could be found in [4,5,7,15,21,23].…”

Capacitance matrix methods for the Helmholtz equation on general three-dimensional regions. [DPOLE, driver program for HELM3D, which solves Dirichlet problem of bounded 3D region imbedded in unit cube, in CDC ANSI FORTRAN for CDC 6600, CDC 7600, Amdahl 470V/6]

“…Hockney [25], [27], Martin [35], Polozhii [40], Proskurowski [41], [42], [43], Proskurowski and Widlund [44], [45], Shieh [46], [47], [48] and Widlund [57]. We refer to Proskurowski and Widlund [44] for a discussion of this development up to the beginning of 1976.…”