Conformal capacity and polycircular domains

by   Harri Hakula, et al.
Qatar University
Turun yliopisto

We study numerical conformal mapping of multiply connected planar domains with boundaries consisting of unions of finitely many circular arcs, so called polycircular domains. We compute the conformal capacities of condensers defined by polycircular domains. Experimental error estimates are provided for the computed capacity and, when possible, the rate of convergence under refinement of discretisation is analysed. The main ingredients of the computation are two computational methods, on one hand the boundary integral equation method combined with the fast multipole method and on the other hand the hp-FEM method. The results obtained with these two methods agree with high accuracy.



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1 Introduction

During the past century, domain functionals

of planar domains have been extensively studied in geometric function theory and other fields of mathematical analysis. Some of these functionals are harmonic measure, hyperbolic distance and its various generalizations, conformal radius, and various domain characterics, for instance, uniform perfectness. One way to classify these notions is their invariance properties such as invariance under conformal mappings, Möbius transformations or stretchings. We study here one of these functionals, the conformal capacity of a condenser. A

condenser is a pair where is a domain and is a non-empty compact set and the conformal capacity is defined as follows


where is the family of all harmonic functions with for all and when . In concrete applications the sets and have a simple geometry, both have a finite number of components, each component being a piecewise smooth curve. In this paper we assume that the domain is multiply connected and all of its boundary components are finite unions of circular arcs or linear segments. Such domains are known as polycircular domains. This term is due to Crowdy c. In this case, it is known that the infimum is attained by a harmonic function (Ah, p. 65), du. Moreover, this extremal function is a solution to the Dirichlet problem


and hence the capacity can be expressed in terms of this extremal function as


where . Let , , where encloses all the other curves . Then each of the boundary components is a piecewise smooth Jordan curve with a finite number of corner points. We assume that none of these corner points is a cusp. The orientation of the total boundary is assumed to be such that is on the left of .

In their classical book G. Polya and G. Szegö ps studied extensively various estimates for capacities in terms of various domain functionals, however their capacity was not the conformal capacity. V. Dubinin du studied conformal capacity in many function theoretic applications and the book of J. Garnett and D. Marshall garmar is an extensive treatise of harmonic measure and its applications to potential theory and geometric function theory.

In spite of their important role in geometric function theory, explicit formulas or numerical values of conformal invariants are known only in the simplest cases. Roughly speaking one can say that when the domain connectivity increases, the more difficult it is to find formulas. For the condenser capacity case, the connectivity of depends on the number of topological components of and is doubly connected if is a simple curve. Our goal here is to continue our earlier work nrrvwyz; nv1; nv and to develop algorithms for the computation of . For an example, see Figure 1.

Numerical computation of conformal capacity has been studied in some earlier papers nv1; nv, but we have not seen any results for conformal capacities of condensers for multiply connected polycircular domains of the type described above. Due to the conformal invariance of the conformal capacity, an auxiliary step in the computation is often to apply conformal mappings onto a canonical domain to simplify the geometry kuh. The books of N. Papamichael- N. Stylianopoulos ps10 and P. Kythe ky and the long survey of R. Wegmann Weg05 are valuable general sources, see also (ps10, pp.14-16) for an overview of the literature. We will now review the most recent literature from the point of view of polycircular domains.

Conformal mapping of Jordan domains with the boundary consisting of a union of finitely many circular arcs, has been studied by P. Brown and M. Porter bp, p, by U. Bauer and W. Lauf BL, and, in particular, by D. Crowdy c. See also BjG; Ho; Tr. In the case of multiply connected polycircular domains, conformal mapping onto canonical domains have been investigated in ber; gol; koe; neh; tsu. Several types of canonical domains are used as an intermediate step of computation. Some examples are those where the boundary components are circles or parallel slits or concentric circular arcs. D. Crowdy c developed an extensive theory based on Schottky’s prime function method to investigate conformal mapping for multiply connected polycircular domains. The goals of his research were different from ours as our focus here is on conformal capacity. M. Badreddine, T. DeLillo, and S. Sahraei bds compare several numerical conformal mapping methods for multiply connected polycircular domains. The above Dirichlet problem (1.2) in multiply connected domains is a wide area of research, see e.g. ais. Computer graphics applications of conformal mapping of multiply connected domains appear in kyyg.

(a) Map.
(b) Potential.
Figure 1: Example of a polycircular ring domain. The lens boundary is defined by two circular arcs passing through the points , respectively. Here . The computed capacity is . In (a) the conformal map is shown, and in (b) the potential function is illustrated.

The quantity of interest, the capacity of a condenser, is directly applicable in engineering contexts. Since the Dirichlet problem is one of the primary numerical model problems, any standard solution technique can be viewed as validated. Here the novelty is in the verification of the results. Instead of merely comparing different discretizations, we make use of two different methods to verify the correctness of the results. The first method is based on the Boundary Integral Equation (BIE) with the generalized Neumann kernel as developed and implemented in MATLAB in a series of papers during the past two decades, see e.g. Nas-Siam1-Nas-PlgCir. The method uses the fast multipole method implementation from Gre-Gim12 for the speed-up of solving linear equations with a special structure. The BIE method has been used to solve several problems in domains of very high connectivity, domains with piecewise smooth boundaries, domains with close-to-touching boundaries, and in domains of real-world problems. In a recent series of papers (nrv2-nv), the method was applied for the capacity computation of planar condensers and for the analysis of isoperimetric problems for capacity. In particular, we will make use of the very recent results nv1.

The second method, -FEM, is based on a Mathematica implementation developed and widely tested by H. Hakula during the past two decades, HRV; HRV2. The method allows one to incorporate a priori knowledge of the singularities into highly non-quasiuniform meshes, within which the polynomial order can vary from element to element. In the class of problems discussed here, one would expect exponential convergence in the natural norm if the discretisation is refined properly.

The BIE method converges exponentially for analytic boundaries and algebraically for piecewise smooth boundaries, see e.g., Nas-ETNA. However, so far, no numerical comparison with other methods has been made in the literature, especially for domains with corners. One of the objectives of this paper is to present such a comparison where the BIE is compared against the -FEM. We obtain surprisingly good agreements between the two methods. The obtained results show that the BIE indeed gives accurate results for domains with corners even if the angles at the corners are small.

The structure of this paper is as follows. In Section 2, we introduce some notation and terminology. Section 3 is a short summary of the two methods we use, the

-FEM and the boundary integral equation method. In Section 4 we compute the capacities of several multiply connected polycircular condensers and compare the performance of the methods. Moreover, the computational error is analyzed as a function of the number of degrees of freedom. Section 5 deals with condensers

where the compact set has lens-shaped structure and we study how closely the theoretical Möbius invariance can be observed in numerical computations. Some concluding remarks are given in Section 6.

2 Preliminary notions

The capacity of a condenser defined in the introduction (1.1) can be defined in terms of the Dirichlet problem (1.2) as well as in many other equivalent ways as shown in du, GMP. First, the family functions in (1.1) may be replaced by several other families by (GMP, Lemma 5.21, p. 161). Furthermore, the capacity is equal to the modulus of a curve family


where is the family of all curves joining with the boundary in the domain and stands for the modulus of a curve family (GMP, Thm 5.23, p. 164). For the basic facts about capacities and moduli, the reader is referred to du; GMP; HKV.


A Jordan domain in the complex plane is a domain with boundary homeomorphic to the unit circle. A quadrilateral is a Jordan domain together with four distinguished points which define a positive orientation of the boundary. In other words, if we traverse the boundary, then the points occur in the order of indices and the domain is on the left hand side. The quadrilateral is denoted by The modulus of the quadrilateral (ps10, p. 52) is a unique positive number such that can be conformally mapped by some conformal map onto the rectangle with vertices with

The modulus is denoted The following basic formula is often used:


A rectangle with sides and and its vertices define a quadrilateral. Depending on the labeling of its vertices, its modulus is either or An alternative equivalent definition of the modulus of a quadrilateral is based on the Dirichlet-Neumann problem L.V. Ahlfors (Ah, Thm 4.5, p. 65).