3.6 Implementation aspects
3.6.2 Information transfer between non-nested meshes
As indicated before, we keep the transfer concept still abstract in this section. The imple- mentation of concrete prolongation and restriction operators is discussed, in particular, in Section 5.7. However, let us remark that the routines for the information transfer between non-nested meshes are usually built from local coupling contributions such as global or local integrals or function evaluations. This holds true whether or not the transfer is actually local in the sense of Definition 3.8.
As the set of non-nested meshes (T`)`=0,...,L does not come with natural parent–child
relations stemming from a regular refinement procedure, one needs to compute suitable neighborhood relations between elements in successive meshes to ensure that the eventual assembly routines of the prolongation matrices are local. For this purpose, we have incor- porated the quadtree/octree implementation of [5] into obslib++. Suitable advancing front techniques exploiting the connectivities of the single meshes can be applied instead; see, e. g., [89] in a related context. Although a hierarchical structure is an adequate choice to treat general problems in a flexible fashion, note that a plain sorting variant may be more efficient for certain cases in the present context. This holds particularly true if the elements of the used meshes are distributed rather evenly because in this case the overhead of the hierarchical quadtree/octree structure might not be negligible.
Naturally, any reasonable choice of coarse meshes satisfiesPL
`=0|T`| . |TL|. This implies
that the number of operations to compute the desired relations of the elements in the pairs (T`−1, T`)`=1,...,L(or in the pairs (T`, TL)`=0,...,L−1if the variant setupSGMGimmis employed
in case of the additive semi-geometric preconditioners) grows at most like O(nLlog nL). In
Section 5.7, it turns out that we achieve optimal complexity O(nL) for the actual assembly
of the sequence of prolongation matrices, having these relations ready to hand.
Coarse spaces from non-nested coarse meshes
Our practical implementation is indeed as flexible as the theoretical considerations in this chapter indicate. The coarse meshes can be unstructured; they do not need to be nested. The user provides the fine level mesh TLwith respect to which the boundary value problem
to be solved is actually set. In addition, coarse level meshes (T`)`=0,...,L−1 are imported
using the extended geometry handling incorporated into obslib++ by [76, 100], too. We have implemented a new module nnmglib in obslib++ to manage the setup of the semi-geometric (monotone) multigrid methods and additive preconditioners. This library also includes the methods for the computation of a variety of concrete transfer concepts (see Chapter 5 and in particular Section 5.7) as well as the methods for the elaborate study concerning the information transfer between non-nested meshes as such (see Section 5.8). Let us point out that, at an intermediate stage, a part of the basic data structures, which
3.6 Implementation aspects 67
are derived from related transfer classes in ug, has been developed during the preparation of a student research project [76].
Note that the mesh TLis directly equipped with a set of degrees of freedom as the finite
element space VL= XLand a basis ΛLare known from the start. In contrast, the nodes and
the elements of the meshes (T`)`=0,...,L−1 merely represent auxiliary geometric entities and
are thus not yet included in the algebraic structure. Their intended purpose is to supply the bases (Λ`)`=0,...,L−1 which are not used in the eventual semi-geometric algorithm. In other
words, we proceed as customary in algebraic multigrid methods; the coarse level degrees of freedom are not created before the setup.
As indicated in the discussion of the algorithms in Section 3.2.2 and Section 3.2.3, once the discrete operators (Π``−1)`=1,...,L or (ΠL`)`=0,...,L−1 and (A`)`=0,...,L−1 are known,
i. e., once the matrices (Π``−1)`=1,...,Lor (ΠL`)`=0,...,L−1 and (A`)`=0,...,L−1 are computed by
setupSGMG or setupSGMGimm, the presented multilevel iterations are nothing but alge-
braic operations involving those matrices. In principle, one does not need the prolongation matrices to be given explicitly. It is sufficient to have routines performing the respective evaluations for given residual or correction vectors ready to hand.
The efficiency of the multigrid method relies on the effectivity of the individual smooth- ing iterations. More precisely, at each level ` ∈ {1, . . . , L}, one needs to be able to reduce the oscillating error components with respect to the space X` sufficiently fast, i. e., using very
few (Gauß–Seidel) iterations. The remaining error has to be sufficiently smooth, namely its representation at the coarser level in X`−1 is sufficiently accurate. In standard geometric
multigrid methods, one relies on sequences of nested meshes with h`−1 = 2h`. For our
purposes, the coarse meshes need to be chosen appropriately such that a similar coarsening assumption holds. Conversely, an increased number of local relaxations in certain regions, possibly carried out in a special order or as block relaxation, could compensate for a “locally bad choice” of the coarser mesh. There are indeed first approaches, although not in the present context, which try to control or optimize the amount of local work performed by the smoother during the algorithm. One example is an a priori redistribution of the total number of relaxation steps towards regions with badly shaped elements. We learned about this methodology from [90].
As a matter of fact, for the above conditions on an effective interplay of smoothing and coarse level correction to be satisfied in practice, one only needs to guarantee that the coarsening factor is in a rather generous range. Our numerical examples include results on the basic robustness of the semi-geometric approach with respect to the choice of the coarse meshes. However, for adaptively refined meshes, we cannot eliminate the possibility that an undesirable local relation between coarse and fine mesh affects the convergence behavior, unless a robust method for readjusting the coarse mesh is applied. In case a highly non-uniform mesh originates from an adaptive refinement procedure (presumably based on suitable error estimators), one might exploit this additional information for an adaptation of the coarse level meshes. A careful elaboration of these issues is beyond the scope of this thesis, though. The utility of an automatic coarse mesh construction, in other words of coarsening procedures, is discussed in Section 4.2.
4
Other geometry-based multilevel
techniques
In the past years, several methodologies have been developed for the application of basic multilevel algorithms to problems with complicated boundaries of the computational do- main. In this chapter, we describe some of the accomplishments of the research on multigrid methods since the very first algorithms have been recorded with respect to finite difference schemes in the unit square. Some of the efforts which have been made to improve the applicability of general multilevel ideas are explained; relevant connections to the introduc- tion of the semi-geometric framework of the previous chapter are established. In part, the presented approaches aim at constructing coarse approximations of finite element spaces as- sociated with unstructured meshes. Others employ structured meshes coming with tailored discretizations which allow for a (to some extent) straightforward coarse level hierarchy.
Certainly, most of the developments to be reviewed in this chapter are in a sense inter- woven. One cannot overlook that the research activities of the cited authors have influenced each other in some form or another. Some ideas immediately build upon the theoretical and algorithmical achievements in multigrid and domain decomposition methods presented in Chapter 2, whereas it seems that others have been developed from a somewhat different point of view.
In agreement with the overall concept of this thesis, we focus on geometric techniques and discuss some important methodologies. Multigrid methods based on adjusted dis- cretizations, which are mostly built from structured meshes, are reviewed in Section 4.1. Next, in Section 4.2, we turn our attention to geometric coarsening techniques for unstruc- tured meshes. We always point out relevant connections and draw comparisons.
As we prefer to put our emphasis on a most thorough study of the properties of the semi- geometric framework, it is beyond the scope of this thesis to evaluate all of the algorithmic ideas described below in detail. We believe that the present chapter provides an adequately deep insight into the development of multilevel methods for problems with complicated boundaries, though. Moreover, we select one method working with a special discretization and present a (monotone) multigrid method based on parametric finite elements in Chap- ter 7. Note that the overall structure of the paradigm put forward in Chapter 3 is a rather general one compared with other concepts which have been investigated in this or a related context.
4.1
Geometric multigrid methods with
adjusted discretizations
In this section, we report on several geometric multilevel techniques based on adjustments of the (fine level) discretizations. Each of the specified approaches employs a discretization scheme of a special nature to construct a suitable coarse space hierarchy. So, for the methods in this section, one has to be willing to give up some of the advantages that finite element discretizations associated with unstructured meshes have. We have also considered another
70 4 Other geometry-based multilevel techniques
way to adjust the discretization for the above purpose, namely a parametric finite element approach. A brief study of a (monotone) multigrid method based on a parametric concept is provided as an excursus in Chapter 7.
The goal of our presentation is to highlight structural similarities and differences and relate the methodologies to the semi-geometric framework of Chapter 3 and to each other. In part, the basic idea is not at all difficult and, e. g., immediately builds on Cartesian (auxiliary) meshes. Both the analysis and the algorithmic realization may be more involved, though. A major difference of the approaches to be described in this section, compared to the semi-geometric concept, is the fact that a new discretization of the problems set up in Chapter 1 is introduced. The formulation of the respective finite element spaces is designed to allow for natural coarse scales. However, just as before, one may need fine level information to evaluate coarse level functions, too.
In this thesis, we do not consider meshfree discretizations such as the partition of unity or generalized finite element methods. Instead of using finite element spaces associated with proper meshes, such an approach basically employs a partition of unity associated with an overlapping decomposition of the computational domain and local approximation spaces; see [10, 145] and, for a more recent overview, [11]. Multilevel methods for partition of unity discretizations of elliptic partial differential equations have been developed, e. g., in [99, 170]. We refer to [55, 62] for recent analytical results.
Before going into detail about the single approaches, we consider it particularly impor- tant to point out that here the relation of two successive meshes is generally much closer than in the semi-geometric setting; the families of meshes exhibit some additional struc- ture. This will be easy to see for the methods described in Section 4.1.1, Section 4.1.2 and Section 4.1.4, but it especially holds true for the composite finite element method re- viewed in Section 4.1.3. Still, the latter is indeed a “multilevel method based on non-nested meshes”, too. The research of this and the other techniques has been driven by the demand to construct multilevel finite element splittings for efficient iterative solvers. In contrast to the more flexible semi-geometric framework, the techniques considered here cannot be applied to given unstructured meshes. We emphasize that, among all studied methods, the variant of composite finite element spaces described in Section 4.1.4 is the only one which is able to resolve the domain without changing the gradients of the multilevel finite element functions. Unfortunately, the straightforward applicability is affected by the disadvantage that the Dirichlet boundary conditions require special attention.
As in the previous chapter, auxiliary spaces and the spaces which are finally used in the multilevel algorithms are denoted by X and V , respectively, with some indices.