article Free AccessNumerical Solution of the Neumann and Mixed Boundary Value Problems by Boundary Contraction Authors: Tse-Sun Chow Boeing Airplane Company, Renton, Washington Boeing Airplane Company, Renton, WashingtonView Profile , Harold Willis Milnes General Motors Corporation, Warren, Michigan General Motors Corporation, Warren, MichiganView Profile Authors Info & Claims Journal of the ACMVolume 8Issue 3July 1961 pp 336–358https://doi.org/10.1145/321075.321078Published:01 July 1961Publication History 6citation612DownloadsMetricsTotal Citations6Total Downloads612Last 12 Months21Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

In this paper the numerical solution of Laplace's equation for the circle is discussed and consideration is given to the convergence of the solution obtained by the boundary contraction method to the analytic solution. It is proved that in order to achieve this a relation between the mesh sizes in the circumferential and radial direction must exist. It is also demonstrated that the error due to the contraction of boundaries can be made insignificant.

A numerical method is introduced for solution of the Dirichlet problem within a circle. The data prescribed on the original boundary is used to compute the solution at mesh points on a concentric circle of slightly smaller radius, thereby defining another boundary value problem over the smaller circle. The solution for the entire region is obtained by successive contractions of the boundary in this way. Appropriate approximating relations are presented and their stability is discussed.