Reflection operators and domain decomposition methods in transport theory problems

New results related to the theory and numerical solution of problems of particle transport in slab geometry are presented in this paper. The results are obtained on the existence of traces of functions in the spaces which are widely used in transport theory. The necessary and sufficient conditions are derived for the solvability of the problems investigated. Estimates are obtained for the norms of boundary values of the solutions and for those of reflection operators. Several iterative algorithms based on the domain decomposition method are constructed and their convergence rate is estimated. The development of parallel computers for solving various problems of mathematical physics has stimulated intensive development of the domain decomposition method [e.g. 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]. It was shown that the investigation of these algorithms can be reduced to that of special-type operators, namely, reflection operators, which were first introduced in model astrophysical problems in [6] and later investigated in [1,8,9,14,15]. On the other hand, the investigation of a number of properties of reflection operators can be reduced to analysing the problem of the existence of traces of functions and the aspects related to boundary value problems. Investigations of this kind were carried out by the author in [3,4] on the problems of particle transport in three-dimensional (3D) geometry. New results related to the theory and numerical solution of problems of particle transport in slab geometry are presented herein and the problem of the existence of traces of functions in the spaces which are widely used in transport theory is also solved. The results are employed to formulate the necessary and sufficient conditions for solvability of the problems discussed. Estimates are obtained for the norms of boundary values of the solutions; these estimates are later used to prove highestaccuracy estimates for the norms of reflection operators. Several iterative algorithms based on the domain decomposition method are constructed and their convergence rate is estimated. 1. NOTATION, ASSUMPTIONS AND FORMULATION OF THE PROBLEM In many of the problems of monoenergetic transport theory, the domain in which the process of particle transport is considered can be represented as a slab {x = (x1,x2,x^): — oo