The Convergence of the Cascadic Conjugate-Gradient Method under a Deficient Regularity

Abstract

In this paper, we deal with a cascadic conjugate-gradient method (shortly called CCG-algorithm). This algorithm is a simpler kind of multigrid (multilevel) methods. We define it recurrently for discrete symmetric positive-definite problems on a sequence of grids. At the coarsest grid, the linear system is solved directly. At the finer grid, the system is solved iteratively by the conjugate-gradient method. A starting guess is an interpolation of the approximate solution from the previous grid. We do not implement any preconditioning or restriction onto a coarser grid. Nevertheless, CCG-algorithm has the same optimal property as the multigrid method. Namely, this algorithm converges with a rate which is independent of an amount of unknowns and a number of grids. In [6], this property was proved for a two-dimensional elliptic second-order Dirichlet problem in a convex polygon Ω, where a solution u belongs to space H ²(Ω). Here we study the problem in a non-convex polygon Ω, where u ∉ H ²(Ω) due to a strong growth of second derivatives near some angular points. We prove the rate of convergence in two cases: for uniform grids and for grids with special refinement near some angular points. The paper [1] contains impressive numerical examples both for usual and deficient regularity.

Research was supported by Institut für Angewandte Analysis und Stochastik im Forschungsverbund Berlin e.V. and by Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg.