Convergence Analysis of All-at-Once Multigrid Methods for Elliptic Control Problems under Partial Elliptic Regularity

Abstract

In this paper we consider the convergence theory for an all-at-once multigrid method for a distributed optimal control problem. Such an analysis has been recently done [J. Schöberl, R. Simon, and W. Zulehner, SIAM J. Numer. Anal., 49 (2011), pp. 1482--1503]. Here, we give a new proof which is based on a more straightforward approach. The main benefit of this new approach is the possibility to extend the analysis to domains where full elliptic regularity, i.e., $H^2$-regularity for the Poisson problem, cannot be guaranteed.