Predictor-corrector domain decomposition algorithm for parabolic problems on graphs

Abstract

In this paper, we present a predictor-corrector type algorithm for solution of linear parabolic problems on graph structure. The graph decomposition is done by dividing some edges and therefore we get a set of problems on sub-graphs, which can be solved efficiently in parallel. The convergence analysis is done by using the energy estimates. It is proved that the proposed finite-difference scheme is unconditionally stable but the predictor step error gives only conditional approximation. In the second part of the paper it is shown that the presented algorithm can be written as Douglas type scheme, based on the domain decomposition method. For a simple case of one dimensional parabolic problem, the convergence analysis is done by using results from [P. Vabishchevich. A substracturing domain decomposition scheme for unsteady problems. Comp. Meth. Appl. Math. 11(2):241-268, 2011]. The optimality of asymptotical error estimates is investigated. Results of computational experiments are presented.