Formulation of the iterative process on subdomains for transport theory problems in odd P2N+1-approximation

Abstract

An iterative process is formulated on subdomains for odd P 2N +i-approximation in a bounded domain K c jR 3 conditionally decomposed into a finite set of subdomains. The convergence of the iterative algorithm is proved. This paper is a logical sequel to [2] where a similar problem was solved for a large class of boundary value problems with a monoenergetic integro-differential transport operator.Let V c £ 3 be a bounded domain in which we seek for the solution to a boundary value problem in P 2 N+i approximation [3], conditionally decomposed into a finite number of subdomains K v , v = 1,2,..., v 0 . The iterative algorithm suggested although somewhat similar to the Schwartz subdomain alternating method differs from the latter first of all in the fact that the domains V v are adjacent without overlapping. In our algorithm (in the Schwartz algorithm as well) it is possible in principle to reduce the problem in a geometrically complex domain V to the subsequent solution of problems in geometrically simplified subdomains V v .The paper is generally aimed at proving the convergence of the iterative process on the basis of the generalized solution concept.

Fictitious components method and the modified difference counterpart of the Schwartz method

“…We conclude with considering the variational difference counterpart of the subdomain alternating method proposed in [7]. As applied to solving system (3.2), this method can be presented in the following form: Ί.Γ«Π_Γ7 JLH'L^( [3][4] where «°e£ w , ; = 0,1,..., the non-singular /cth-order matrix S 0 is determined by the bilinear form L hSh (u,v) in (1.2).…”

“…Similar ideas were suggested in [2], and on the differential level in [7,8]. The major results have been obtained for systems of mesh equations which approximate the Dirichlet problem for the two-dimensional Helmholtz equation by the finite element method [6] using piecewise-linear prolongations.…”

Grid-projection method with small angles in the cells

“…The consistency conditions analogous to (2.2), (2.3) are used in the Schwarz alternation method [8][9][10][11]18].…”

“…By a solution of this system we shall mean the values of the unknowns /ju for which the weighted sum of the squares of residuals of these equations is minimal. In order to minimize this functional we use the Schwarz alternation method [8][9][10][11]18], where separate cells of the grid are taken as subdomains.…”

Subdomain iterative method for elliptic-type quasi-linear equations

“…The subdomain iterative method generalizing the classical Schwartz algorithm to the case of adjacent non-overlapping domains was investigated in [1][2][3][4][5][6]. Therein the authors considered boundary value problems for linear elliptic equations, transport theory equations, and also for elasticity theory equations.…”