Symmetrization of the odd P2N + 1-approximation of single-velocity transport equation

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

“…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.…”

“…1. The system of differential equations in P 2N +1 -approximation can be written in the following form [3]: Here, Qe£7 (U is a set of unit vectors in # 3 ); {Υ ( ™\Ω)} is an orthogonal basis of spherical functions^; σ η (χ) are bounded non-negative piecewise-continuous functions°' n = ° (2) σ° > Ο, η ^ 1…”

“…The specific forms of conditions (2) and (3) are not required in this paper; these conditions are described in detail in [3]. Conditions (l)-(3) can be certainly combined, i.e.…”

“…Boundary conditions (l)-(3) make it possible to symmetrize the problem by eliminating functions φ 2Λ +1 m (x) (with the odd first subscript) in system (1). As a result, we realize the variational formulation of the problem [3] with the following functional subject to minimization: (5) …”

“…The set of all functions of form (3) satisfying requirements (9) and subject to [in cases (2) and (3)] the main boundary conditions [3] will be denoted by H. By virtue of the known inclusion theorem [6], on each piece wise-smooth surface Γ of the class C 1 , lying inside the domain V or coinciding with its boundary S, each of functions (9) has a unique trace Φ 2 λ,/η( χ Γ) 6^2 (Π· If using the determined (generalized) solution φ°(χ,Ω)εΗ and the second subsystem in (1) we determine functions Φ 2 λ+ι m( x ) [ an<^ h ence > the function φ 1 (χ,Ω)], the latters belong to L 2 (V) and, therefore, do not have unique traces on Γ. In the concluding part of the paper (see Appendix) we justify the principle of prolongation of φ 1 (χ,Ω) onto Γ related in the natural way to the variational identity…”