In the standard domain decomposition theory the resulting subdomains are often assumed to have a certain regularity, as in Toselli and Widlund (2005, Assumption 4.3), where each subdomain is a finite union of coarse scale elements and the number of coarse elements forming the subdomain are uniformly bounded. This assumption does not always hold. Subdomains might be generated from a mesh partitioner, or be the result of a decomposition scheme with slight or systematic alterations of the subdomain following refinement, e.g. see the type 3 domain in Dohrmann et al. (2008a, figure 5.1) and the snowflake domain in Fig. 1. In this paper we will assume that each subdomain is a connected union of fine scale elements.
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