On the computation of modules of long quadrilaterals

Abstract

We study quadrilaterals Q which are given by two intervals on {~: Im ~ = 0} and {~: Im ~ = 1}, and two Jordan ares 7t, 3'z in {~: 0 < I m ~ < I} connecting these two intervals. Many practical problems require the determination of the module re(Q) of Q, but if Q is "long," i.e., if h ,= min{Re ~: ~ ~72} --max{Re ~: ~ E ~'1} is large, the conformal mapping of Q onto a rectangle becomes difficult because of the crowding effect. However, it turns out that re(Q) -h approaches a limit very quickly, as h ~ oo, and we can therefore estimate re(Q) -[m(Qt) + ra(Q2)] when Q is decomposed into two smaller quadrilaterals Q1, Q2. Several numerical examples are presented. This domain decomposition method goes back to Papamichael and Stylianopoulos [9]. On the other hand, experiments showed that re(Q) behaves rather regularly if Q is expanded horizontally. This caused Papamichael and Stylianopoulos [8], [9] to

A domain decomposition method for approximating the conformal modules of long quadrilaterals

“…In the remainder of this section we state without proof various results which are needed for our subsequent work. These include, in particular, three theorems due to Gaier and Hayman [2,3] which play a crucial role in our work.…”

“…In this paper we consider again the problem of computing approximations to re(Q), and investigate the possibility of extending the application of the DDM to a wider class of quadrilaterals than that studied in [2,3,9,10]. Our main objective is to show that the method does, indeed, have much wider applicability and, for each new application, to derive an estimate of the error in the resulting DDM approximation to re(Q).…”

“…Our main objective is to show that the method does, indeed, have much wider applicability and, for each new application, to derive an estimate of the error in the resulting DDM approximation to re(Q). We shall do this by making use of two new theorems, which are simple consequences of the results of Gaier and Hayman [2,3] mentioned in (iv) and (v) above. The paper is organized as follows: In Sect.…”

“…The paper is organized as follows: In Sect. 2 we set up various notations and state, without proof, the results of [2,3] which are needed for our subsequent work. In Sect.…”

On variational principles of conformal mappings

“…In the case where B is simply connected, Γ is an arc on the boundary of B, and z 0 ∈ (∂B)\Γ, the quantity r(B, Γ, z 0 ) was treated in essence in [15,16]. The study of differences and ratios of Robin capacities is of particular interest; see [7,11] and [17,Chapter 7.2].…”