Fast and Accurate Computation of the Logarithmic Capacity of Compact Sets

Abstract

We present a numerical method for computing the logarithmic capacity of compact subsets of $\mathbb {C}$, which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it.

Appendix: Numerical computation of the preimage of a parallel slit domain

Let $\varOmega$ be a given parallel slit domain, i.e., the entire z-plane with $\ell$ slits $L_j$, $j=1,2,\ldots ,\ell$, along straight lines; see the top of Fig. 8. An efficient numerical method for computing the conformal map $z=\omega (\zeta )$ from an unbounded domain G exterior to $\ell$ smooth Jordan curves $\varGamma _j$, $j=1,2,\ldots ,\ell$, onto the parallel slit domain $\varOmega$ such that $\omega (\zeta ) = \zeta + O(1/\zeta )$ as $\zeta \rightarrow \infty$ has been presented in [17, Sec. 4.5]. Assume that the boundary $\varGamma$ of G is parametrized by the function $\eta (t)$ as in (3.1). Assume also the operators $\mathbf{N}$ and $\mathbf{M}$ are the same operators as in (3.3). Then we have the following theorem from [17].

Theorem A.1

Let

$$\begin{aligned} \gamma (t)={\mathrm {Im}}[\eta (t)], \quad t\in J, \end{aligned}$$

let $\mu$ be the unique solution of the boundary integral equation

$$\begin{aligned} (\mathbf{I}- \mathbf{N}) \mu = - \mathbf{M}\gamma , \end{aligned}$$

(A.1)

and let h be the piecewise constant function

$$\begin{aligned} h = ( \mathbf{M}\mu - (\mathbf{I}- \mathbf{N}) \gamma )/2. \end{aligned}$$

Then the function f with the boundary values

$$\begin{aligned} f(\eta (t))=\gamma (t)+h(t)+\mathrm {i}\mu (t) \end{aligned}$$

is analytic in G with $f(\infty )=0$ and the conformal mapping $\omega$ is given by

$$\begin{aligned} \omega (\zeta )=\zeta -\mathrm {i}f(\zeta ), \quad \zeta \in G\cup \varGamma . \end{aligned}$$

Fig. 8

The given parallel slit domain $\varOmega$ (top), the initial preimage domain $G^0$ (dashed line, bottom), and the computed preimage domain G (solid line, bottom)

In Theorem A.1, the domain G is assumed to be known, and the integral equation (A.1) is used to the find the conformal map $z=\omega (\zeta )$ from G onto the parallel slit domain $\varOmega =\omega (G)$. In our application with the Cantor sets, however, the domain G is unknown and the parallel slit domain $\varOmega$ is known. Hence a straightforward application of a numerical method based on Theorem A.1 is not possible.

We will now describe an iterative method developed in [20] for computing G and the conformal map from G onto the (known) parallel slit domain $\varOmega$. The method is an improvement of a numerical method suggested by Aoyama et al. [2], where the preimage G is assumed to be circular. Since the image region $\varOmega$ is elongated (parallel slit domain), crowding can cause serious problems. Further, the convergence of the iterative method is slow if G is assumed to be circular. To overcome such difficulties, it was assumed in [20] that the boundaries of the domain G are ellipses instead of circles.

Let $|L_j|$ denote the length of the slit $L_j$ and let $z_j$ denote its center, $j=1,2,\ldots ,\ell$. In the iteration step $i=0,1,2,\ldots$ we assume that the domain $G^i$ is a multiply connected domain bounded by the ellipses $\varGamma ^i_j$, parametrized for $j=1,2,\ldots ,\ell$ by

$$\begin{aligned} \eta ^i_j(t)=\zeta ^i_j+0.5\left( a^i_j\cos t-\mathrm {i}b^i_j\sin t\right) , \quad t\in J_j=[0,2\pi ]. \end{aligned}$$

Then the following iteration computes the centers of the ellipses $\zeta ^i_j$, the lengths of the major axes $a^i_j$, and the lengths of the minor axes $b^i_j$ for $j=1,2,\ldots ,\ell$.

Initialization Let $\varepsilon >0$ be a given tolerance and let Max be a maximum number of iterations. (In our numerical experiments in this paper we always used $\varepsilon =10^{-14}$ and $\mathrm{Max}=50$.) Set

$$\begin{aligned} \zeta ^0_j=z_j, \quad a^0_j=(1-0.5r)|L_j|, \quad b^0_j=r a^0_j, \end{aligned}$$

where $0<r<1$ is the ratio of the lengths of the major and minor axes of the ellipse; see Fig. 8 (dashed line, bottom) for $r=0.5$.

For $i=1,2,\ldots$:

• 1. Map $G^{i-1}$ to a parallel slit domain $\varOmega ^i$ (based on Theorem A.1), which is the entire $z-$plane with $\ell$ slits $L^i_j$, $j=1,2,\ldots ,\ell$, along horizontal straight lines.

• 2. If $|L^i_j|$ denotes the length of the slit $L^i_j$ and $z^i_j$ denotes its center, then we define the parameters of the preimage domain $G^i$ as

  $$\begin{aligned} \zeta ^{i}_j&= \zeta ^{i-1}_j-(z^{i}_j-z_j), \\ a^{i}_j&= a^{i-1}_j-(|L^{i}_j| -|L_j|), \\ b^{i}_j&= r a^{i}_j. \end{aligned}$$

• 3. Stop the iteration if

  $$\begin{aligned} \mathop {{ max}}\limits _{1\le j\le m}(|z^{i}_j-z_j|+||L^{i}_j| -|L_j||)<\varepsilon \quad \mathrm{or} \quad i>{Max}. \end{aligned}$$

Several numerical examples in this paper as well as in [2, 20] show the convergence of this iterative method, but no proof of convergence has been given so far. Numerical experiments also show that the iterative method requires fewer iterations for small values of r, i.e., thin ellipses. For thin ellipses, however, we usually need a larger number of points n for discretizing the boundary integral equations and the GMRES method for solving these discretized equations requires more iterations to converge.

In the numerical experiments with the Cantor sets shown in this paper we have not chosen to optimize upon these parameters, but we used the fixed values $r=0.5$ and $n=64$. The number of iterations for the convergence to the accuracy $\varepsilon =10^{-14}$ of the above iterative method applied in the computation of $c(E_k)$ for $k=1,2,\ldots ,12$ is shown in Fig. 9. The (unpreconditioned) GMRES method for solving the discretized integral equations required between 5 and 11 iterations.

Fig. 9

Number of iterations for computing the preimage domains required in the computation of $c(E_k)$ for $k=1,2,\ldots ,12$