The numerical solution of three-dimensional electrostatic field problems in electron-optical devices with complicated boundaries requires the use of very large and fast computers. A numerical counterpart of the Alternating Procedure by H. A. Schwarz (SAP) makes efficient use of both storage space and time available on the computer and therefore represents a useful new method in this context. In this method. the domain of solution is decomposed into independent subdomains where the appropriate individual methods of solution may be chosen. Correspondingly. the original boundary value problem is decomposed into a sequence of boundary value problems in these subdomains. SAP requires no explicit influence matrix and uses little storage space. Time requirements depend essentially on the methods applied in the respective subregions. SAP is particularly useful in obtaining analytical formulas for field distributions in "large" regions of empty space (such as in fringe field regions). Numerical results obtained with a computer program treating an electrostatic quadrupole lens problem show that a high degree of accuracy can be obtained with this method.
Schaefer, Christoph H.
"The Application of the Alternating Procedure of H.A. Schwarz for Computing Three-Dimensional Electrostatic Fields in Electron-Optical Systems,"
Scanning Electron Microscopy: Vol. 3
, Article 8.
Available at: https://digitalcommons.usu.edu/electron/vol3/iss1/8