On the Applicability of Lions' Energy Estimates in the Analysis of Discrete Optimized Schwarz Methods with Cross Points

for k = 1,2, . . . and i = 1, . . . ,n, where Ωi ⊂ Ω are non-overlapping subdomains, Γi j = ∂Ωi ∩Ω j is the interface between Ωi and an adjacent subdomain Ω j, j �= i, and pi j > 0 are Robin parameters along Γi j. In [7], the powerful technique of energy estimates is used to show convergence of (2) for η = 0 under very general settings. Similar techniques have been used to prove convergence results for other types of equations, cf. [2] for the Helmholtz equation and [5] for the time-dependent wave equation. While one often assumes that the proof carries over trivially to finiteelement discretizations, it has been reported in the literature (cf. [9, 8]) that discrete OSMs can diverge when the domain decomposition contains cross points, i.e., when more than two subdomains share a common point. This is in apparent contradiction with Lions’ proof, and such difficulties contribute to the limited use of OSMs in practice. The goal of this paper is to explain why the presence of cross points makes it possible for the discrete OSM to diverge despite the proof of convergence at the continuous level, and why this difference in behavior is generally unavoidable. The remainder of the paper proceeds as follows. In Section 2, we recall Lions’ energy estimate argument. In Section 3, we explain why it is impossible to convert the