Extending the application domain of the model order reduction method in calculating the electrostatic field

A problem of determining the electrostatic field formed by a set of charged electrodes has been considered.  The details of the approximate solving of the Dirichlet problem have been given for the Laplace's equation in a substantially spatial formulation based on the use of the model order reduction method.  The mathematical models have been improved and the problem of calculating the electrostatic field has been simplified, taking into account the present symmetry of electrodes positioning in electronic optics systems.  For the eighth-order abstract group, three independent structures of the corresponding class of systems have been identified.  The application domain of the model order reduction method based on finite-group theory for numerically solving integral equations has been extended by transforming the initial boundary-value problem not containing symmetry groups into two problems.  The boundary surface of one of them has a finite symmetry group and the other allows for a sufficiently simple numerical solution.  This simplification of the problem is aimed at improving the accuracy of computational methods, eliminating sources of instability of these methods, and speeding up computations.  To confirm the efficiency of the proposed algorithm, a model problem of calculating the electrostatic field of a quadrupole lens has been considered.  The example of its solving demonstrates all the advantages of the developed computational algorithm.  A number of numerical experiments have been conducted.  The electrostatic field of the corresponding planar approximations has been calculated to verify the validity of the obtained results.

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Mathematical Modeling and Computing, Vol. 6, No. 2, pp. 358–366 (2019)