Geodesic
Method of decomposition of the computational domain in problems of high-current electronics
Sibirskij žurnal industrialʹnoj matematiki, Tome 18 (2015) no. 2, pp. 124-130.

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The method of decomposition of the computational domain is known from solving linear problems of mathematical physics. In this article, we propose to use this method for calculating intensive beams of charged particles in nonlinear self-consistent problems of high-current electronics. The computational domain partitions into two subdomains: the cathode-full subdomain and the principal subdomain. In the cathode-full subdomain, we construct an analytic solution from known formulas while in the principal subdomain the solution is found numerically. The central question is that of coordinating the subdomains. To this end, on the conjugation condition, by analogy with linear problems, we write down the Poisson–Steklov equation, which is approximated by a system of operator nonlinear equations. It is solved by methods of quasi-Newton type, namely, by Broyden's method. As follows from the experiments, the process converges already at the fourth iteration with precision acceptable for practice.
Keywords: self-consistent problem, Poincaré–Steklov equation, nonlinear equation. 
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V. M. Sveshnikov. Method of decomposition of the computational domain in problems of high-current electronics. Sibirskij žurnal industrialʹnoj matematiki, Tome 18 (2015) no. 2, pp. 124-130. http://geodesic.mathdoc.fr/item/SJIM_2015_18_2_a11/

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