A domain decomposition method for conformal mapping onto a rectangle

Papamichael, N.; Stylianopoulos, Nikos

doi:10.1007/bf01888163

Cited by 18 publications

(23 citation statements)

References 12 publications

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“…For this reason, they are sharper than those given in Theorems 1, 4, 2 and 5 of their earlier paper [2]. [9], without imposing any restrictions on the sizes of h and h2. However, the estimates given in [9] were derived by imposing certain additional assumptions on the arcs Yl, 72, and one of these is rather restrictive.…”

Section: Theorem 22 ([3 Theorem 6] [2 Theorem 2]) For the Decompmentioning

confidence: 65%

“…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).…”

Section: Introductionmentioning

confidence: 97%

“…Such a system is said to be a quadrilateral Q and is denoted by and also for various other errors connected with the DDM approximation to the conformal map Rmte~ ~ f2. These are derived in [9,Sect. 4], by assuming that the two boundary arcs 71, 72, satisfy certain smoothness conditions.…”

Section: Introductionmentioning

confidence: 99%

“…(iv) An estimate for the error in the DDM approximation to re(Q), which (unlike our estimate in [9]) is derived without imposing any conditions on the boundary segments Yl, V2, other than requiring that they are Jordan arcs. (This proves one of the conjectures made in [10].)…”

Section: Introductionmentioning

confidence: 99%

“…(ii) A number of conjectures concerning the assumptions under which the error estimates of [9] hold (cf. [10,Sect.…”

Section: Introductionmentioning

confidence: 99%

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

Papamichael

Stylianopoulos

1992

Numer. Math.

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Summary. This paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals. The method has been studied already by us and also by D. Gaier and W.K. Hayman, but only in connection with a special class of quadrilaterals, viz. quadrilaterals where: (a) the defining domain is bounded by two parallel straight lines and two Jordan arcs, and (b) the four specified boundary points are the four corners where the arcs meet the straight lines.Our main purpose here is to explain how the method may be extended to a wider class of quadrilaterals than that indicated above. Mathematics Subject Classification (1991): 30C30, 65E05

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Section: Theorem 22 ([3 Theorem 6] [2 Theorem 2]) For the Decompmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…(ii) A number of conjectures concerning the assumptions under which the error estimates of [9] hold (cf. [10,Sect.…”

Section: Introductionmentioning

confidence: 99%

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

Papamichael

Stylianopoulos

1992

Numer. Math.

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Conformal mapping of long quadrilaterals and thick doubly connected domains

Laugesen

1994

Constr. Approx

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Abstract. In this paper we investigate theoretically an approximation technique for avoiding the crowding phenomenon in numerical conformal mapping. The method applies to conformal maps from rectangles to "long quadrilaterals," i.e., Jordan domains bounded by two parallel straight lines and two Jordan arcs, where the two arcs are far apart. We require that these maps take the four corners of the rectangle to the four corners of the quadrilateral.Our main theorem tackles a conformal mapping problem for doubly connected domains, and we derive from this our results for quadrilaterals. As a corollary, we extend the "domain decomposition" mapping technique of Papamichael and Stylianopoulos.Similar results hold for the inverse maps, from quadrilaterals to rectangles.

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On the computation of modules of long quadrilaterals

Gaier

Hayman

1991

Constr. Approx

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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

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A domain decomposition method for conformal mapping onto a rectangle

Cited by 18 publications

References 12 publications

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

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

Conformal mapping of long quadrilaterals and thick doubly connected domains

On the computation of modules of long quadrilaterals

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