Computational Methods and Function Theory 4 (2004), No. 2, 475--496
Copyright Heldermann Verlag 2004
Marcus Stiemer
stiemer@math.uni-dortmund.de , Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany.
The Robin capacity $\rob(A)$ of a compact, non-empty set $A\subset\partial\Omega$ with respect to a domain $\Omega\subset\CD$ containing $\infty$ is defined by $$ \rob(A)= \rob(A,\Omega)= \exp\!\left(\lim_{z\to\infty} -R(z)+\log|z|\right), $$ where $R(z) = R(z,\infty)$ is the fundamental solution of a mixed boundary value problem with pole at $\infty$, where Dirichlet conditions are imposed on $A$ and Neumann conditions on $\partial\Omega\setminus A$. P.\ Duren and M.\ Schiffer have discovered that it coincides with the minimal logarithmic capacity of $f(A)$ over all conformal mappings $f$ of $\Omega$ with $f(z) = z+\LandauO(1)$, $z\to\infty$. In this article, effective methods for the numerical determination of $\rob(A)$ are developed. For this purpose the conformal invariant $\rob(A)/\kap(\partial\Omega)$ is related to other moduli of the given configuration like harmonic measure or conformal modulus. Then an effective extremal point discretization for these moduli based on Menke points is derived. If $\Omega$ is analytically bounded, the discretizations presented provide geometrically fast converging approximations to the considered moduli and thus to Robin capacity.
Keywords: Robin capacity, conformal invariants, extremal points.
MSC 2000: Primary 30C85; Secondary 30E10, 31A15.
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