The boundary value problem for discrete analytic functions

Handle URI:
http://hdl.handle.net/10754/562780
Title:
The boundary value problem for discrete analytic functions
Authors:
Skopenkov, Mikhail
Abstract:
This paper is on further development of discrete complex analysis introduced by R.Isaacs, J.Ferrand, R.Duffin, and C.Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S.Smirnov from 2010. This was proved earlier by R.Courant-K.Friedrichs-H.Lewy and L.Lusternik for square lattices, by D.Chelkak-S.Smirnov and implicitly by P.G.Ciarlet-P.-A.Raviart for rhombic lattices.In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A.Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory. © 2013 Elsevier Ltd.
KAUST Department:
Visual Computing Center (VCC)
Publisher:
Elsevier BV
Journal:
Advances in Mathematics
Issue Date:
Jun-2013
DOI:
10.1016/j.aim.2013.03.002
Type:
Article
ISSN:
00018708
Sponsors:
The author was partially supported by "Dynasty" foundation, by the Simons-IUM fellowship, and by the President of the Russian Federation grant MK-3965.2012.1.
Appears in Collections:
Articles; Visual Computing Center (VCC)

Full metadata record

DC FieldValue Language
dc.contributor.authorSkopenkov, Mikhailen
dc.date.accessioned2015-08-03T11:05:26Zen
dc.date.available2015-08-03T11:05:26Zen
dc.date.issued2013-06en
dc.identifier.issn00018708en
dc.identifier.doi10.1016/j.aim.2013.03.002en
dc.identifier.urihttp://hdl.handle.net/10754/562780en
dc.description.abstractThis paper is on further development of discrete complex analysis introduced by R.Isaacs, J.Ferrand, R.Duffin, and C.Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S.Smirnov from 2010. This was proved earlier by R.Courant-K.Friedrichs-H.Lewy and L.Lusternik for square lattices, by D.Chelkak-S.Smirnov and implicitly by P.G.Ciarlet-P.-A.Raviart for rhombic lattices.In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A.Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory. © 2013 Elsevier Ltd.en
dc.description.sponsorshipThe author was partially supported by "Dynasty" foundation, by the Simons-IUM fellowship, and by the President of the Russian Federation grant MK-3965.2012.1.en
dc.publisherElsevier BVen
dc.subjectAlternating currenten
dc.subjectBoundary value problemen
dc.subjectDiscrete analytic functionen
dc.subjectEnergyen
dc.titleThe boundary value problem for discrete analytic functionsen
dc.typeArticleen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.identifier.journalAdvances in Mathematicsen
dc.contributor.institutionInstitute for Information Transmission Problems of the Russian Academy of Sciences, Bolshoy Karetny per.19, bld.1, Moscow, 127994, Russian Federationen
kaust.authorSkopenkov, Mikhailen
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