Convergence of the multigrid $V$-cycle algorithm for second-order boundary value problems without full elliptic regularity

Abstract

The multigrid V-cycle algorithm using the Richardson relaxation scheme as the smoother is studied in this paper. For second-order elliptic boundary value problems, the contraction number of the V-cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps, without assuming full elliptic regularity. As a consequence, the V-cycle convergence result of Braess and Hackbusch is generalized to problems without full elliptic regularity.