Schwarz methods
This is a convenient point to comment briefly on the application of the above concept to two-level overlapping Schwarz methods with a global coarse space. For a detailed introduc- tion into this field, see, e. g., [177, Chapter 3]. We also mention the enormous number of references in this monograph. The idea of this class of methods is to decompose the com- putational domain into overlapping subdomains, preferably of simple structure, on which local (e. g., exact) solvers can be successfully applied. The information transfer between the single problems is realized by the local overlaps and a global component.
Without going into great detail, we point out some significant connections to the previ- ous considerations. The notations are a little different here. As there are only two distinct levels, the fine and coarse quantities may be labeled with the indicesh and H, respectively.
In this context, let us mark a decomposition into subspaces by the index i (instead of `).
Assume that V = Xh is a given finite element space associated with a mesh Th of the
computational domain Ωh = Ω to approximate the solution as described in Section 1.3. Let
the local finite element spaces
(Vi)i=1,...,N with Vi⊂ V, i ∈ {1, . . . , N },
be associated with a “horizontal” overlapping decomposition (Ωi)i=1,...,N of Ω. In particular,
3.5 A coarse space for overlapping Schwarz methods 63
but only to the local spaces (Vi)i=1,...,N in practice, which are defined with respect to local
meshes (Ti)i=1,...,N. As most proof techniques involve a coloring argument, one usually
assumes that the number of adjacent subdomains is bounded (independently of h and N ). A Schwarz preconditioner, which acts as (successive or parallel) subspace correction method corresponding to the above decomposition, is in general not scalable with respect to the number of subdomains N . To prove preconditioning results independent of the number of subdomains, one usually introduces a global coarse space; see, e. g., [177]. For a historical overview of the role of coarse spaces in domain decomposition methods, we refer to [189]. In many cases, bounds on the condition number of the preconditioned operator may be proved, which essentially depend on the ratio of the maximum size of the subdomains to the minimum size of the overlaps. Coarse spaces associated with elementary coarse meshes are ready to hand as long as the structure of the decomposition is simple. To achieve the mentioned result, in general, one needs to ensure enough global information transfer by choosing the coarse mesh size sufficiently small, e. g., comparable to the size of the subdomains. In addition, an (almost) exact solution of the subproblems is often required.
Two-level overlapping Schwarz methods have also been developed for problems with unstructured meshes. In this case, as we have seen in the multilevel setting, it may be difficult to construct nested coarse spaces as the local fine meshes do typically not allow for a proper global coarse mesh. Advanced coarse spaces for this problem class include partition of unity spaces [160, 161, 162] and spaces obtained by smoothed aggregation [134, 164]. A paper on recent progress of the theory of domain decomposition methods with irregular subdomains is [188]; see also the references therein. Here, we focus on an approach which is close to the one presented before. If XH is a global finite element space
associated with a mesh TH of ΩH ⊃ Ω which is not related to the space Xh, a coarse space
can be constructed by means of a suitable prolongation operator ΠhH : XH → Xh correlating
the global coarse mesh with the local fine meshes. Similarly to the multilevel case, a nested space is defined via
V0 := ΠhHXH (3.23)
like in [43, 49, 51, 52]. Then, the analysis of the decomposition
V = V0+ N
X
i=1
Vi,
namely the proof of a partition lemma, requires respective H1-stability and L2-approxi- mation properties of the applied operator ΠhH. This may indicate that the considerations of the present chapter and the research of geometrically inspired transfer concepts in Chapter 5 are useful for other purposes, too.
Naturally, in this two-level setting, we do not need to worry about the fact that a composed mapping is used in the proof; the composition always consists of as few as two mappings. This means that the stability and approximation results for the respective fine- to-coarse mapping QV
0 : V → V0 are optimal if the coarse space is constructed via (3.23)
and if the operator ΠhH satisfies the meantime well-known properties.
In the context of the semi-geometric multilevel methods, not only for the analysis but also for practical purposes, the locality of the information transfer is crucial. Here, the
64 3 Semi-geometric multilevel preconditioners
situation is a little different. Whereas, in a multilevel setting, the reduction of the number of the degrees of freedom per level is only by a factor typically in the order of 2−d, one may be able to manage with very few coarse degrees of freedom per subdomain in case of the overlapping two-level method. Therefore, it might not be mandatory to use a local transfer operator. A global transfer concept might not affect the overall efficiency as only small dense systems need to be solved corresponding to the subspace V0. Admittedly, this does
not hold true for the asymptotic range as the size of the subdomains and, thus, the coarse mesh size needs to decrease to retain spaces (Vi)i=1,...,N with sufficiently small dimension.