The principle of iteration over subdomains in problems involving the transport equation

Reflection operators and domain decomposition methods in transport theory problems

“…10) is the optimal one only if R iterations are carried out. However, after R iterations the algorithm can be continued so that it is stable for a certain sequence {^}/°i 0 (Α<Α|<ΑΙ+Ι, I-+OO, /-κχ>) and is again optimal at the R r th iteration [18]. and #!…”

“…2,10,11,17]. In [18] this method was suggested for transport theory problems. The iterative algorithms based on the domain decomposition method, with an optimal choice of parameters, were constructed in [14,15].…”

Grid-projection solution of an elliptic problem for an irregular grid

“…If the boundary of the /-th cell coincides with the domain boundary, then boundary conditions are imposed at the appropriate point. When applying the alternation Schwarz method, it has been shown in [23][24][25] that third-kind conditions like those in (1.8a) or (1.8b) yield the best convergence. We used these conditions initially in the iterative process, when collocation-type conditions were considered.…”

“…To solve the system of equations (1.12), (1.13) for / = I,...,/, where / is the total number of the grid nodes, we applied an iterative method, which is similar to the Schwarz alternation method (see, for example, [12,[14][15][16][17][23][24][25][26]). The method used can be viewed as a block Gauss-Seidel method with convergence acceleration of iterations [21,22].…”

“…The Schwarz alternation method [12,[14][15][16][17][23][24][25][26] was used as an iterative method to solve the resultant system of linear algebraic equations. A separate cell was the subdomain in which the solution was sought in the form of a quadratic polynomial.…”