Conformal capacity and polycircular domains

Hakula, Harri; Nasser, Mohamed M. S.; Вуоринен, Матти

doi:10.1016/j.cam.2022.114802

Cited by 2 publications

(1 citation statement)

References 56 publications

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“…Table 4.2 shows the absolute differences between the computed values, which indicates a good agreement between the two methods. As in [14], the values computed using the FEM will be considered as reference values and used to estimate the error in the values computed by the BIE method for several n. The BIE method cannot be used for d = 0. The error for d = 0.05, 0.1, .…”

Section: Verification Of Results Let Us Consider the Case With Four D...mentioning

confidence: 99%

Mobile disks in hyperbolic space and minimization of conformal capacity

Hakula,

Nasser,

Vuorinen

2024

etna

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Our focus is to study constellations of disjoint disks in the hyperbolic space, i.e., the unit disk equipped with the hyperbolic metric. Each constellation corresponds to a set E which is the union of m > 2 disks with hyperbolic radii r j > 0, j = 1, . . . , m. The centers of the disks are not fixed, and hence individual disks of the constellation are allowed to move under the constraints that they do not overlap and their hyperbolic radii remain invariant. Our main objective is to find computational lower bounds for the conformal capacity of a given constellation. The capacity depends on the centers and radii in a very complicated way even in the simplest cases when m = 3 or m = 4. In the absence of analytic methods, our work is based on numerical simulations using two different numerical methods, the boundary integral equation method and the hp-FEM method, respectively. Our simulations combine capacity computation with minimization methods and produce extremal cases where the disks of the constellation are grouped next to each other. This resembles the behavior of animal colonies minimizing heat flow in arctic areas.

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Section: Verification Of Results Let Us Consider the Case With Four D...mentioning

confidence: 99%

Mobile disks in hyperbolic space and minimization of conformal capacity

Hakula,

Nasser,

Vuorinen

2024

etna

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Estimating conformal capacity using asymptotic matching

Miyoshi

Crowdy

2023

IMA Journal of Applied Mathematics

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Conformal capacity is a mathematical quantity relevant to a wide range of physical and mathematical problems and there has been a recent resurgence of interest in devising new methods for its computation. In this paper we show how ideas from matched asymptotics can be used to derive estimates for conformal capacity. The formulas derived here are explicit, and evidence is given that they provide excellent approximations to the exact capacity values even well outside the expected range of validity.

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Conformal capacity and polycircular domains

Cited by 2 publications

References 56 publications

Mobile disks in hyperbolic space and minimization of conformal capacity

Mobile disks in hyperbolic space and minimization of conformal capacity

Estimating conformal capacity using asymptotic matching

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