1 Introduction
In this article, we analyze scaling properties of the Schwarz overlapping domain decomposition method for the Poisson problem: find such that
Here is a bounded Lipschitz domain and denotes the usual Sobolev space of functions with weak derivatives in with vanishing Dirichlettrace. We investigate the behavior of the Schwarz iterative method when consists of a increasing number of fixedsize overlapping subdomains . We are particularly interested in the case that the subdomains are overlapping balls with comparable radii.
The motivation for studying this problem comes from numerical simulations in computational chemistry. Recently, a domain decomposition method has been propose [1, 16, 18, 19, 22] in the context of socalled implicit solvation models, more precisely for the COnductorlike Screening MOdel (COSMO) [14] which is a particular type of continuum solvation model (CSM). In a nutshell, such models account for the mutual polarization between a solvent, described by an infinite continuum, and a charge distribution of a given solute molecule of interest. It therefore takes the longrange polarization response of the environment (solvent) into account. We refer to the review articles [17, 24] for a thorough introduction to continuum solvation models.
While for most of the applications of domain decomposition methods, the computational domain remains fixed (such as in engineeringlike applications) and finer and finer meshes are considered, applications in the present context deal with different molecules consisting of a (very) large number of atoms. Each atom is associated with a corresponding van der Waals (vdW)ball with a given and elementspecific radius so that the total computational domain consists of the union of those vdWballs. For a set of different molecules the computational domain is therefore changing and the Schwarz domain decomposition exhibits different convergence properties. For example, for an (artificial) linear chain of atoms of increasing length the Schwarz domain decomposition is scalable and does not require a socalled coarse space correction.
A general convergence analysis of the Schwarz domain decomposition iterative method for the family of domains , , can not easily be deduced from classical analyses of domain decomposition methods available in the literature, e.g. [8, 25, 26]. This is due to the fact that these classical analyses assume a fixed domain that is decomposed in an increasing number of (overlapping) subdomains of decreasing size, whereas in the setting outlined above the subdomains all have a given comparable size and the global domain changes when the number of subdomains is increased. It turns out that for a convergence analysis in the latter case it is not obvious how results and tools available in classical analyses can be applied. Therefore, in recent papers [2, 3, 4, 5, 6] this topic has been addressed and new results on the convergence of the Schwarz domain decomposition iterative method on a family of domains , , were obtained. More precisely, the first theoretical results were obtained for a chain to rectangles in two dimensions [2, 3], which were later generalized to chainlike structures of disks and balls in two respectively three dimensions [4, 5]. These results, however, cover only (very) special cases as each of the subdomains has a nonempty intersection with the boundary of the computational domain , i.e. no balls are allowed that are contained inside . A first step towards a more general analysis can be found in [6], which analyzes how the error propagates and contracts in the maximum norm in a general geometry. It is shown that for a molecule with layers, it takes iterations until the first contraction in the maximum norm is obtained. This is essential to understand the contraction mechanism, in particular for the first iterations, but unfortunately does not provide much insight on the rate of (asymptotic) convergence.
In this paper we present a general analysis which covers many cases that occur in applications and that goes beyond the previously mentioned contributions. Although the presentation is somewhat technical, due to the fact that we have to formalize the geometry of the family of domains , , the convergence analysis is based on a few fundamental ingredients known from the field of subspace correction methods and Sobolev spaces, which are combined with new descriptors of the geometries considered. We outline the main components of the analysis. We use the wellestablished framework of subspace correction methods [26]. In [27], for the successive (also called “multiplicative”) variant of the Schwarz domain decomposition method a convergence analysis in a general Hilbert space setting is derived. The contraction number of the error propagation operator (in the natural energy norm) can be expressed in only one stability parameter ( in Lemma 2.1 below). This parameter quantifies the stability of the space decomposition. We bound this stability parameter by introducing a new variant of the pointwise Hardy inequality
. This variant allows estimates that take certain important global geometry properties into account. Using this we derive, for example, a uniform bound in
, if we have a chain like family of domains, and a bound that grows (in a specified way) as a function of , if we have a family of “globular ” domains. It is wellknown from the literature on domain decomposition methods that in the latter case one should use an additional “global coarse level space”. We propose such a space for our setting and analyze the rate of convergence of the Schwarz method that includes this additional coarse space.The paper is organized as follows. In Section 2 the Schwarz domain decomposition that we analyze in this paper is explained and an important result on the rate of convergence of this method, known from the literature, is given. This result essentially states that the contraction number (in the energy norm) of the Schwarz method is characterized by only one quantity, which controls the stability of the space decomposition. In Section 3 we introduce new descriptors of the specific class of domains (union of overlapping balls) that is relevant for our applications. Furthermore, for this class of domains a natural partition of union is defined and analyzed. This partition of unity is used in Section 4 to derive bounds for the stability quantity. A further key ingredient in our analysis of the Schwarz method is a variant of the pointwise Hardy inequality, that is also presented in Section 4. A main result of the paper is given Theorem 4.1. As is wellknown from the theory of Schwarz domain decomposition methods, in certain situations the efficiency of such a method can be significantly improved by using a global (coarse level) space. For our particular application this issue is studied in Section 5. Finally, in Section 6 we present results of numerical experiments, which illustrate certain properties of the Schwarz domain decomposition method and relate these to the results of the convergence analysis.
2 Problem formulation and Schwarz domain decomposition method
We first describe the class of domains that we consider. Let , , be the centers of balls and the corresponding radii. We define
We consider the Poisson equation: determine such that
(1) 
with a given source term . For a subdomain we denote the Sobolev seminorm of first derivatives by . For solving this problem we use the Schwarz domain decomposition method, also called successive subspace correction in the framework of Xu et al. [27]. This method is as follows:
(2)  
endfor 
This is a linear iterative method and its error propagation operator is denoted by . We then obtain
An analysis of this method is presented in an abstract Hilbert space framework in [27]. Here we consider only the case, in which the subspace problems in (2) are solved exactly.
Remark 2.1
There also is an additive variant of this subspace correction method (called “parallel supspace correction” in [27]). This method may be of interest because it has much better parallelization properties. This additive variant can be analyzed with tools very similar to the successive one given above. We further discuss this in Remark 5.2.
As norm on it is convenient to use . In this norm the bilinear form has ellipticity and continuity constants both equal to 1. The corresponding operator norm on is also denoted by . We need the following projection operator defined by
We recall an important result from [27] (Corollary 4.3 in [27]). Assume that is closed in . Define
(3) 
where for all . Then
(4) 
holds.
The constant quantifies the stability of the decomposition of the space into the sum of subspaces . Due to the result (4) we have that the contraction number of the Schwarz domain decomposition method (in the natural norm) depends only on . Hence, if is independent of certain parameters (e.g., in our setting ) then the contraction rate is also robust w.r.t. these parameters. In the remainder of this paper we analyze this stability quantity depending on the geometrical setting and the closedness assumption needed in Lemma 2.1. The analysis is based on a particular decomposition with and forms a partition of unity that is introduced and analyzed in the next section.
3 Geometric properties of the domain and partition of unity
The number of balls is arbitrary and in the analysis below it is important that in estimates and in further results we explicitly address the dependence on the number . The estimates in the analysis below depend on certain geometry related quantities that we introduce in this section.
In order to formalize the geometry dependence in our estimates, we introduce a (infinite but countable) family of geometries indexed by the increasing number of balls, where each element represents the set of balls defining the geometry characterized by the set of centers and radii. We further introduce
We first start with stating basic assumptions on the geometric structure of the considered domains.
Assumption 3.1 (Geometry assumptions)

[leftmargin=1cm,style=nextline,itemsep=5pt,font=,topsep=3pt]
 (A1)

For each , we assume that is connected. This assumption is made without loss of generality. If has multiple components, the problem (1) decouples into independent problems on each of these components and the analysis presented below applies to the problem on each component.
 (A2)

For each , we assume that there are no , with , such that , i.e., balls are not completely contained in larger ones. Otherwise, the inner balls can be removed from the geometric description without further consequences.
 (A3)

We assume the radii of the balls to be uniformly bounded in the family : there exists and such that
 (A4)

(Exterior cone condition) We assume that for each there exists a circular cone with positive angle , apex and axis that belongs entirely to the outside of in a neighborhood of , i.e., for sufficiently small. We furthermore assume that is uniformly bounded from below: there exists such that
Related to this we have the following result. For denote by , , all indices such that and define . Assumption (A4) is equivalent to the following one:
There exists such that for each and for each
, there exists a unit vector
such that for all . This implies that all vectors are situated on one side of the plane that is perpendicular to and passing through . The limiting angle of a cone with apex in the direction of is given by the minimal angle of with the tangential plane at to each ball , , illustrated by in Figure 1. Note that . Hence is bounded away from zero if and only if is bounded away from , i.e. bounded away form zero. Finally note that , which shows that the uniform boundedness away from zero of the interior cone angles and of are equivalent conditions.The condition of Lemma 3.1 provides a precise mathematical statement in terms of geometrical notions. This condition for instance excludes the following scenarios, using the notation , and where Figure 1 (middle and right) provides a schematic illustration of those two cases:
 :

Intersection of two balls in only one point, that is, and the centers are aligned on one line. In turn, only the plane passing through which is perpendicular to line passing through and does not intersect locally around and there exists no cone of positive angle wit apex that (locally) belongs to the outside of .
 :

Intersection of three balls in one point. Here, only the line , with being the normal vector to the plane passing through , , and , belongs to the outside of . In turn, there exists no cone of positive angle wit apex that (locally) belongs to the outside of .
3.1 Local geometry indicators
We introduce certain geometry descriptors, which we call indicators, that are related to the specific geometry of the domain and that will be used in the estimates derived below. We will distinguish between local and global indicators, the former only being dependent on local geometrical features whereas the latter being dependent on the global topology of the geometric configuration.
We introduce some further definitions. We take a fixed , with corresponding domain . We decompose the index set into two disjoint sets by introducing (“interior balls”) and (“boundary balls”). The corresponding (overlapping) subdomains are denoted by , . It may be that is an empty set. We define, for , , . For we define . Further, define
Hence, on the function is the distance function to , which is extended by 0 outside . Note that for and that, for any , there holds that if and only if .
In the following, we list the indicators that are used in the upcoming analysis.
Indicator 3.1 (Maximal number of neighbors)
Define
(5) 
i.e., is the maximal number of neighboring balls that overlap any given ball .
Indicator 3.2 (Maximal overlap indicator)
Let be the smallest integer such that:
(6) 
Hence, is the maximal number of neighboring balls that overlap any given point of any given ball .
Indicator 3.3 (Stable overlap indicator; interior balls)
Note that for we have , hence for all , and thus for all . Thus, it follows that there exists such that for all . We thus define and there holds
(7) 
Note that by construction, this is a local indicator.
Remark 3.1
The indicator is a measure for the amount of overlap between any interior ball and its neighboring balls. The indicator is small if there exists a point , with , that is simultaneously close to and to the boundary of all spheres with .
A proof of the following lemma, giving rise to a further indicator, is given in Appendix 8.1. [Stable overlap for boundary balls] Under Assumption (A4), there exists , such that
(8) 
Indicator 3.4 (Stable overlap for boundary balls)
The constant defined in Lemma 3.1 is considered as a geometry indicator.
The four indicators introduced above are all natural ones, which are directly related to the number of neighboring balls and the size of the overlap between neighboring balls.
We need one further local indicator, which needs some introduction. In the analysis of the Schwarz method we use a (natural) partition of unity, cf. Section 3.3. The gradient of some of these partition of unity functions is unbounded at , where their growth behaves like . To be able to handle this singular behavior, we need an integral Hardy estimate of the form
cf. Corollary 4.1. One established technique to derive such an estimate is as in e.g. [11, 13], where pointwise Hardy estimates are used to derive integral Hardy estimates. The analysis in this approach is based on a certain “fatness assumption” for the complement of the domain . Here we follow this approach and below we will introduce a local indicator that quantifies this exterior fatness of the domain, which is very similar to the fatness indicator used in [11, 13] (cf., for example, Proposition 1 in [11]). Before we define the fatness indicator, note that due to the definition of as a union of balls and assumption (A4), we have the following property
(9) 
Indicator 3.5 (Local exterior fatness indicator)
Remark 3.2
We call this a local exterior fatness indicator because depends only on a small neighbourhood of , consisting of points that have distance at most to . The quantity essentially (only) depends on two geometric parameters related to exterior cones with apex at , namely the maximum possible aperture (angle of the cone) and the maximal cone height such that the cone is completely contained in . For points lying only on one sphere , the cone can be chosen to be as wide as a flat plane. For points lying on an intersection arc , the largest angle of the cone is determined only by the center and radii of the two balls . Finally any point lying on an intersecting point of three or more boundary spheres can be assigned a cone whose maximal angle depends on the radii and centers of the associated balls. Figure 2 (left) provides a schematic illustration. The maximal cone height at is related to the width of at in the direction of the axes of the cone, see also Figure 2 (right) for a schematic 2Dillustration. If these apertures and heights are bounded away from zero (uniformly in ), the quantity is bounded away from zero. In our applications we consider domains such that the apertures and heights satisfy this property.
The set of local geometry indicators is denoted by
(12) 
We emphasize that depends only on local geometry properties of the domain (as explained above) and does not depend on the global topology of (e.g., not relevant whether is a linear chain of balls or has a globular form for example). We thus assume the following assumption.
Assumption 3.2 (Asymptotic geometry assumption)

[leftmargin=1cm,style=nextline,itemsep=5pt,font=,topsep=3pt]
 (A5)

We assume that the local geometry indicators are uniformly bounded in the family .
3.2 Global geometry indicator
In the analysis below we need a PoincaréFriedrichs inequality for , cf. Lemma 4.3. As is wellknown, the constant in this inequality depends on global geometry properties of the domain and is directly related to the smallest Laplace eigenvalue in . To control this constant we use an approach, presented in Section 4.1 below, based on pointwise Hardy estimates. For this approach to work one needs a measure for “global exterior fatness”. This measure resembles the one used in Indicator 3.5, but there are two important differences. Firstly, we now consider instead of only . Secondly, for , instead of the corresponding closest point projection (used in Indicator 3.5) we now take a possibly different exterior point such that with the exterior volume is comparable to the volume . The latter property is a key ingredient in the derivation of satisfactory pointwise Hardy estimates. Simple examples show that taking is not satisfactory. Below, we present a construction of “reasonable” points that is adapted to the special class of domains that we consider.
In the field of applications that we consider, the notions of the Solvent Accessible surface (SAS) and the Solvent Excluded Surface (SES) are often used and are natural to use in our context. The SES was introduced by Lee and Richardson [15, 21]. The SES is also called the smooth molecular surface or the Connolly surface, due to Connolly’s fundamental work [7]. Explanations and a more mathematical description can also be found in [20]. Here we report only a few key notions and properties that are relevant for our analysis.
For a given general (probe) radius the SAS is defined as
and . Let be the signed distance function to SAS (positive in ), hence, . We define
We refer to Figure 3 for a graphical illustration of the definition of the SES and the SAS. Further, we denote the maximal distance to the socalled Van der Waals surface by
A property of the SASSES construction is that the balls with center on and radius (these are tangent to ) are completely contained in . We will use these balls in the construction of balls , , , that have “sufficient exterior volume”. To determine a suitable we use , with a parameter that will be specified below. We now explain this construction. A closest point projection on and the corresponding distance are denoted by
The maximum distance to is denoted by . Note that the following properties hold
(13) 
Define for all , , with and .
First, note that by the construction of the , there holds with , i.e. this corresponds to the original definition of the SES of “rolling a probe sphere of radius over the van der Waalscavity”. The point denotes the center of probe and lies on the SAS. See Figure 4 for an illustration and [20] for further explanations. Since , there holds that for all
and thus
Second, for any , there holds
and thus
Hence, for we have
We define
(14) 
and thus the exterior fatness estimate for all holds. In the Hardy estimates used below (cf. (24)) we are interested in (posssibly small) bounds of the quotient . This motivates the introduction of the following indicator:
Indicator 3.6 (Global exterior fatness indicator)
Define
(15) 
and the corresponding global fatness indicator
(16) 
It follows that for any , there exists a corresponding point such that
(17) 
Remark 3.3
The function in (15) is not necessarily continuous. Discontinuities can appear, for example, for values at which holes in “disappear”. This function is, however, continuous on , where the are points in at which discontinuities appear, and has on each interval a continuous extension to . From these properties it follows that a minimizer of exists (but may be nonunique).
In the literature, e.g. [13], a property as in (17) is called a uniform (exterior) fatness property of the corresponding domain, and this notion is related to that of variational capacity, hence the name that we have chosen. We continue with a short discussion of this global indicator.

[leftmargin=20pt,style=nextline,itemsep=5pt,font=,topsep=3pt]
 1)

It is clear from its definition that is a global indicator and that it has the following upper and lower bounds
The quantity can be seen as a measure for the globularity of the domain that involves the maximal distance to a () and the maximal distance to the boundary ().
 2)

In the particular case where consists of a linear chain of overlapping uniform spheres of length , we have , , and thus , i.e., is independent of . On the other hand, when considering a geometry of uniform spheres whose centers lie on the unit grid with radius (in order that no inner holes appear), we obtain , , hence, . In this case is proportional to as increases.
 3)

Another consequence of the construction above is that the entire cavity can be covered by balls with centers on the , radii and each of these balls contains a smaller ball of radius that lies entirely in .
3.3 Partition of unity
In this section we introduce a (natural) partition of unity, based on the local distance functions , , and derive smoothness properties for the elements in this partion of unity. In the bounds for derivatives that are proven below only local geometry indicators from are involved.
Note that since is the distance function to the boundary in we have
(18) 
One easily checks that if and only if . We further define
The system forms a partition of unity (PU) of subordinate to the cover :
Remark 3.4
The functions are in general not smooth. In Figure 5 a twodimensional case is illustrated, in which consists of three intersecting disks (in this case we have ). As a further, more precise, illustration we consider the threedimensional case of two overlapping balls , , with . We thus have , , .
The intersection of with is the circle . The functions do not have a continuous extension to the intersection circle . Take an accumulation point and a sequence with . We then have . On the other hand, we can take a sequence
Comments
There are no comments yet.