The convergence of the cascadic conjugate-gradient method applied to elliptic problems in domains with re-entrant corners

Abstract:

We study the convergence properties of the cascadic conjugate-gradient method (CCG-method), which can be considered as a multilevel method without coarse-grid correction. Nevertheless, the CCG-method converges with a rate that is independent of the number of unknowns and the number of grid levels. We prove this property for two-dimensional elliptic second-order Dirichlet problems in a polygonal domain with an interior angle greater than $\pi$. For piecewise linear finite elements we construct special nested triangulations that satisfy the conditions of a “triangulation of type $(h,\gamma ,L)$” in the sense of I. Babuška, R. B. Kellogg and J. Pitkäranta. In this way we can guarantee both the same order of accuracy in the energy norm of the discrete solution and the same convergence rate of the CCG-method as in the case of quasiuniform triangulations of a convex polygonal domain.