On the convergence of an optimal Additive Schwarz method for parallel adaptive finite elements

Abstract In this paper, we formulate and analyze a one-level Additive Schwarz preconditioner with full-domain overlap for a class of parallel adaptive finite elements, including the Bank–Holst paradigm and the local and parallel algorithms based on two-grid discretizations. We show that the generous overlap is more than enough to compensate for the single-level nature of the preconditioner, and the effective condition number of the associated preconditioned operator can be bounded from above independently of the number of the subdomains, the fine and coarse mesh sizes. The main results of this paper substantially extend those in Loisel and Nguyen (2017), in the sense that we are able to remove from the bound of the effective condition number the explicit term, 1 ∕ ( 1 − γ 2 ) , where γ is the maximum of the constants in the strengthened Cauchy–Buniakowskii–Schwarz inequalities applied to the hierarchical decompositions of full-domain local adaptive finite element spaces into the coarse space and fine spaces. In other words, our new results confirm that the proposed preconditioner is even more robust as the constant γ can be close to 1 for adaptive meshes with bad shape regularity. Numerical results are provided to confirm our theoretical finding.