Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The boundary value problem for discrete analytic functions   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.

23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph