Continuous Analysis of the Additive Schwarz Method: a Stable Decomposition in H1 with Explicit Constants

The classical convergence result for the additive Schwarz preconditioner with coarse grid is based on a stable decomposition. The result holds for discrete versions of the Schwarz preconditioner, and states that the preconditioned operator has a uniformly bounded condition number that depends only on the number of colors of the domain decomposition, the ratio between the average diameter of the subdomains and the overlap width, and on the shape regularity of the domain decomposition. The Schwarz method was however defined at the continuous level, and similarly, the additive Schwarz preconditioner can also be defined at the continuous level. We present in this paper a continuous analysis of the additive Schwarz preconditioned operator with coarse grid in two dimensions. We show that the classical condition number estimate also holds for the continuous formulation, and as in the discrete case, the result is based on a stable decomposition, but now of the Sobolev space H1. The advantage of such a continuous result is that it is independent of the type of fine grid discretization, and thus does the more natural continuous formulation of the Schwarz method justice. The upper bound we provide for the classical condition number is also explicit, which gives us the quantitative dependence of the upper bound on the shape regularity of the domain decomposition.