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Discrete Riemann–Hilbert problems, interpolation of simply closed curves, and numerical conformal mapping
Ist Teil von
Journal of computational and applied mathematics, 1988-09, Vol.23 (3), p.323-352
Ort / Verlag
Elsevier B.V
Erscheinungsjahr
1988
Link zum Volltext
Quelle
Elsevier ScienceDirect Journals
Beschreibungen/Notizen
Let
G be a simply connected region with 0 ϵ
G and with a twice Lipschitz continuously differentiable boundary curve, Γ, and let
z
μ, μ = 1,…,
N, be an even number of
N = 2
n equidistant grid points on the unit circle {sfnc
zsfnc = 1} with
z
1 = 1. Then there exists for all sufficiently large
N a polynomial
P̂
n
of degree
n + 1, normalized by the condition that the coefficient
p
0 = 0, and the coefficients
p
1 and
p
n+1
are real, such that
P̂
n
satisfies the interpolation condition
P̂
n
(
z
μ) ϵ Γ for all μ = 1,…,
N. In a neighbourhood of the normalized conformal mapping function
Φ
there is exactly one such interpolating polynomial. The sequence of these
P̂
n
converges to the conformal mapping function
Φ
as
n → ∞. If Γ is three times Lipschitz continuously differentiable, then the
P̂
n
are also conformal mappings of the unit circle onto regions, which approximate
G.
An important tool for the theoretical investigation is a discrete analogon of the Riemann—Hilbert problem. We present two fast procedures for the numerical solution of the discrete Riemann—Hilbert problem: A conjugate gradient method with computational cost O(
N log
N) and a Toeplitz matrix method with cost O(
N log
2
N). Using this, one can calculate the interpolating polynomials by a Newton method and in this way obtains very effective methods for the numerical approximation of the conformal mapping function.