Fast poisson solvers on general two dimensional regions for the Dirichlet problem

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

“…Only numerical schemes of first order accuracy for the interior Neumann problem of the Poisson equation on bounded two dimensional regions with smooth boundaries are considered. See [28] for a similar treatment of the Dirichlet problem. A brief survey of related previous work is given in Section 7 of this work.…”

A capacitance matrix method for Dirichlet problem on polygon region

“…Recently many methods have been developed for the solution of such systems. These methods are based on different techniques, see the papers [2,3,7,9,11,12], and the references given there. The algorithm described here is based on the capacitance matrix technique, see [3,11].…”

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“…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].…”

“…We also note that one of the arguments given in a similar context in Proskurowski and Widlund [44] is incorrect. The proof given above can be modified to give rather crude, but still quite useful estimates of the condition number of the matrix CD, see Shieh [48].…”