L. I. Mochurad, P. Ya. Pukach


The electron-optical systems are basic components of the modern research complexes by the help of which complicated physical processes related to motion of the charged particles in the corresponding potential fieldsare are studied. The real electron-optical systems have a great number of the charged electrodes of the complicated configuration. Therefore, the use of economic collocation method under the conditions of cobbed-permanent approximation of the sought density requires the numeral solving the systems of linear algebraic equations of large dimensions with the densely filled matrices. In the one model example, the а posteriori method of error evaluation and parallelization of the procedures for a class of problems of electronic optics is considered. It is also considered that the connected open surface where boundary conditions are set obtains the Abelian group of symmetry of the sixteenth order. Such problems arise in the mathematical modeling of electronic optics systems. The general method based on integral equation method was improved. Specificity of the problem was taken into account by use of group theory apparatus. This article shows how using the apparatus of the group theory it is possible to solve an initial problem by the help of the sequence of the sixteen independent integral equations, where the integration is realized only on one of the congruent constituents of the surface. It creates the conditions for parallel processes of problem solution in general. The procedure of parallelization was realized with the help of the most popular means of OpenMP. The collocation method for obtaining approximate values of needed "density of charge distribution" in the particular two-dimensional integral equations is used. To take into account the singular way of solving the problem in the circuit of the open surface the a posteriori method of error evaluation is created. The account of specificity of the open-circuit surfaces allows to decrease the amount of the controlled special points considerably and in the best case to deal only with one. It also substantially simplifies the algorithm of calculations. To prove the reliability and estimation of the technique efficiency the number of numerical experiments is carried out. For the representation of the electrostatic field equipotential lines are used, thus, solutions are analysed by the help of distribution of lines of even potential.


mathematical modeling; abelian group of symmetry; integral equations; multi-core processors; software OpenMP

Full Text:



Carstensen, C., Maischak, M., Praetorius, D., & Stephan, E. P. (2004). Residual-based a posteriori error estimate for hypersingular equation on surfaces. Numerische Mathematik, 97(3), 397–425.

Eriksson, K., Estep, D., Hansbo, P., & Johnson, C. (1995). Introduction to Adaptive Methods for Differential Equations. Acta Numerica, 4, 105–158.

Garasym, Y. S., & Ostudin, B. A. (2003). On numerical appoach to solve some three-dimensional boundary value problems in potential theory based on integral equation method. Zhurn. obchysl. ta prykl. matematyky, 1(88), 17–28.

Hlumova, M. V. (2000). Chyselne modeliuvannia fizychnykh protsesiv u visesymetrychnykh elektronno-promenevykh pryladakh. Avtoreferat na zdobuttia naukovoho stupenia kand. fiz.-mat. nauk, Kharkiv, 17 p. [in Ukrainian].

Mochurad, L. (2013). Rozparalelennia protsedur chyselnoho rozviazuvannia zadach ploskoi elektrostatyky, yaki maiut abelevi hrupy symetrii skinchennykh poriadkiv. Visn. Lv. un-tu. Ser. prykl. matem. ta inform, 20, 34–41. [in Ukrainian].

Mochurad, L., Harasym, Y., & Ostudin, B. (2009). Maximal Using of Specifics of Some Boundary Problems in Potential Theory After Their Numerical Analysis. International Journal of Computing, 8(2), 149–156.

Morrison, J. A.,& Lewis, J. A. (1976). Charge Singularity at the Corner of a Flat Plate. SIAM J. Appl. Math., 31(2), 233–250.

Serre, J.-P. (1977). Linear Representations of Finite Groups (Graduate Texts in Mathematics) (v. 42) 1st ed. 1977. Corr. 5th printing 1996 Edition. Springer-Verlag: New-York-Heidelberg-Berlin.

Sybil, Yu. M. (1997). Three dimensional elliptic boundary value problems for an open Lipschitz surface. Matem. studii, 8(2), 79–96.

Voss, M. J. (Ed.). (2003). OpenMP share memory parallel programming. Toronto, Canada, 270 p.

Zakharov, E. V., Safronov, S. I., & Tarasov, R. P. (1992). Abelevy gruppy konechnogo poriadka v chislennom analize lineinykh kraevykh zadach teorii potentciala. Zhurn. vychisl. matematiki i matem. fiziki, 32(1), 40–58. [in Russian].



  • There are currently no refbacks.