Realization and convergence proof of cascadic methods for domains with curvilinear boundary or angular

points

Vladimir Shaidurov

Krasnoyarsk, Computing Center of Russian Academy of Sciences

Abstract

We study a cascadic organization of iterative processes which gives a simpler kind of multigrid methods without any preconditioning or restriction onto a coarser grid. We begin the cascadic algorithm at the coarsest grid where the linear system is solved directly. At finer grids the systems are solved iteratively; a starting guess is an interpolation of the approximate solution from the previous grid. We prove this algorithm to converge with an optimal rate independent of a number of unknowns and a number of grids.

Initially, cascadic conjugate-gradient algorithm was presented by P.Deuflhard. In his and some later papers, Dirichlet problem was treated for elliptic second-order equations on polygon. Here we treat this problem in a domain with curvilinear boundary where one can not ensure a triangulation nestedness. Nevertheless, we use a nested sequence of trial spaces which has analogous properties like one on nested triangulations.

Becides, we treat problems with singularities arising due to angles of boundary. We use special triangulations condenced in the vicinities of angles and prove both the standard rate of iterative convergence and the standard order of accuracy like those in regular case.