Harri Hakula,
Mohamed M. S. Nasser,
Matti Vuorinen
S. 1 - 19
doi:
10.1553/etna_vol60s1
Verlag der Österreichischen Akademie der Wissenschaften
doi:
10.1553/etna_vol60s1
Abstract:
Our focus is to study constellations of disjoint disks in the hyperbolicspace, i.e., the unit disk equipped with the hyperbolic metric. Each constellationcorresponds to a set $E$ which is the union of $m>2$ disks withhyperbolic radii $r_j>0, j=1,\ldots,m$. The centers of the disks are notfixed, and hence individual disks of the constellation are allowed tomove under the constraints that they do not overlap and theirhyperbolic radii remain invariant. Our main objective is to findcomputational lower bounds for the conformal capacity of a givenconstellation. The capacity depends on the centers and radii in a verycomplicated way even in the simplest cases when $m=3$ or $m=4$. In theabsence of analytic methods, our work is based on numerical simulationsusing two different numerical methods, the boundary integral equationmethod and the $hp$-FEM method, respectively. Our simulations combine capacitycomputation with minimization methods and produce extremal cases wherethe disks of the constellation are grouped next to each other. Thisresembles the behavior of animal colonies minimizingheat flow in arctic areas.
multiply connected domains, hyperbolic geometry, capacity computation
Published Online:
2024/01/18 08:55:17
Object Identifier:
0xc1aa5576 0x003ec2db
Rights:All rights reserved.For questions regarding copyright and copies please contact us by email.
Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.
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