Subdomain iteration principle in transport equation problems

Abstract

An iterative process similar to the Schwartz method for non-overlapping adjacent domains is constructed for solving the monoenergetic transport equation in a domain decomposed into a finite number of disjoint subdomains. Under the boundary conditions, most important for applications, the sufficient conditions are determined for the convergence of the process constructed. The paper is mainly aimed at proving the convergence of the iterative method.Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 6/16/15 1:40 PM L J Wi X <^ J and €{^λί f σ(χ)(/ί(χ,Ω)) 2 αχαΩ] 1/2 <Α 1 ||/ΐ||^ι LJ|Fix(l/\Wi) Jwhere λ ι <1 is the estimating constant in the following inequality: 0(χ,ΩΩ')αΩ' = 1 -0(χ,ΩΩ'^Ω' ^ l-goco^ = λ\ < 1. J t/\O)i J £0i

Subdomain iterative method for elliptic-type quasi-linear equations

“…Similarly to [4] we can show that the generalized solution to (1.1), (1.2) with notation (1.12) taken into account satisfies the following relation in each of the subdomains:…”

“…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.…”

“…In this paper, following the technique contained in [4] we prove the convergence of the subdomain iterative method in the case of the elliptic-type quasi-linear equation in an axially symmetric domain.…”

“…and satisfy the condition The convergence of the iterative process can be proved as in [4] on the basis of the generalized solution notion. The function u e W^Qj) is said to be the generalized solution to problem (1.1), (1.2) if any function η e W^(Q) satisfies the identity …”

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

“…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.…”