2.2. ELEMENTOS BÁSICOS TEÓRICOS
2.2.3. Clasificación de Eutanasia
and boundary conditions
In this section, we explain the semi-geometric multilevel hierarchy in more detail, com- plementing the introductory remarks from Section 3.1. The treatment of boundary values for the coarse level problems is also reconsidered. Here, the focus lies on an illustrative presentation and useful practical considerations. We take a more analytic viewpoint next. For the development of the multilevel preconditioning framework, we have chosen to use a quite general set of non-nested meshes. The desire to detach the generation of coarse meshes (and spaces) as far as possible has several reasons. First of all, it allows for employing independent mesh generation processes, possibly of different nature, at each level. The coarse domains (Ω`)`<L can be considerably different from the fine domain ΩL. This may
be beneficial, for example, to acquire coarse meshes of simpler shapes, perhaps even meshes with some regular structures. In other cases, if the computational domain ΩL varies due
to (pseudo-)time stepping or changes slightly during (shape) optimization, it may be of practical use to reuse the coarse level meshes. Finally, we emphasize that the analysis of the presented multilevel methods, which is carried out in the next section, does not require very restrictive assumptions on the particular interaction of two successive meshes. Of course, the case that the coarse meshes are nested is included.
To retain the capability to capture the behavior of the functions under consideration in the whole computational domain, each domain in the sequence (Ω`)`=0,...,L should be
covered by all other domains which are used to provide coarse level information. For the methods with a recursive structure of the information transfer, this is reflected in (3.1).
3.3 Coarse representation of boundaries and boundary conditions 51
For the additive method with immediate mappings (ΠL`)`=0,...,L−1, this condition becomes
weaker as it is sufficient to assume Ω`⊃ ΩL for ` ∈ {0, . . . , L − 1}.
It is important to note that the constructed coarse level spaces resolve the boundary of the computational domain in a certain sense. This is an immediate consequence of the fact that the bases (eΛ`)`=0,...,L−1 are defined by linear combinations of basis functions in ΛL.
For quite a few transfer concepts, each basis eΛ` is a partition of unity in Ω` as asserted in
Lemma 3.2. In any case, the equality
Ω = int[{x ∈ Rd| ∃ v ∈ V`, v(x) 6= 0} holds true for all ` ∈ {0, . . . , L}.
As the domains (Ω`)`=0,...,L are principally independent of each other, the Dirichlet
boundary ΓD ⊂ ∂ΩL of the mixed boundary value problem is assumed to be resolved by
the finest mesh TL only. Still, by the assumption that the range of the operator ΠLL−1 is in
the space XL ⊂ HD1(ΩL), the Dirichlet conditions are incorporated into all coarse spaces
in a very natural way. As a general rule, as we only study the variational approach in this thesis, namely the operators at the coarser levels are entirely defined by a Galerkin relation, the coarse space problems do not need any special considerations of the respective boundaries. This means that all possible boundary conditions only have to be treated in the finest space XL. In particular, the concept used to obtain the matrices (Π``−1)`=1,...,L
may be rather general; no boundary modifications of the employed prolongation operators are necessary, in contrast to [171].
This paradigm (to consider coarse level basis functions as linear combinations of fine level basis funtions and then cancel undesired contributions) reminds of monotone multi- grid methods for variational inequalities. There, the coarse level spaces are modified by a recursive truncation of basis functions depending on the active constraints of the current fine level iterate. This approach ensures that the coarse level correction does not violate the active constraints; see [121, 122, 124]. We explain this in a little more detail in Section 6.2. The resolution of the boundary at coarser levels by design, namely by means of suitable lin- ear combinations of fine level functions, also resembles the composite finite element method [105, 106]; see Section 4.1.3.
For most transfer concepts, especially for the ones which yield “local” operators accord- ing to Definition 3.8 to be considered in the next section, it is reasonable to neglect the elements at level ` which lie completely outside of Ω`+1, ` ∈ {0, . . . , L−1}. Otherwise, if the
domain Ω` constituted by T` is considerably larger than ΩL, it can happen that (eλ`q)q∈N` is not a basis but merely a spanning set of the constructed subspace V`. In this case, the
discrete representation of the composed operator ΠLL−1· · · Π`+1` has zero columns. Indeed, one may also realize this procedure as a modification of the nodes depending on the transfer operators. Assuming that the matrices (Π`+1` )`=0,...,L−1 have been computed according to
some formula to be specified later, we can reduce the sets of nodes via N` 7−→ {p ∈ N` | ∃ q ∈ N`+1, (Π`+1` )pq6= 0}
for descending ` ∈ {0, . . . , L − 1}. Naturally, in a practical implementation the redundant degrees of freedom associated with neglected nodes are never created. The above procedure
52 3 Semi-geometric multilevel preconditioners
needs to be included in the setup of the semi-geometric multilevel hierarchy setupSGMG (Algorithm 3.3).
Finally, let us remark that it is possible to slightly relax the conditions on the coarse domains such that they only need to cover the interior nodes. For the Dirichlet part of the boundary, this can be achieved without additional assumptions on the transfer operators. However, for the Neumann boundary, one needs to construct special interpolation operators [47, 91] because plain extension by zero to the part of the fine domain lying outside of the coarse domain is not sufficiently accurate. We postpone the technical discussion to the end of Section 3.4.3; see Remark 3.11 and Remark 3.13.