On the numerical approximation of unstable
minimal surfaces with polygonal boundaries |
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Authors: | Michael Hinze |
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Institution: | Technical University of Berlin,
Fachbereich Mathematik,
Stra?e des 17. Juni 136, D-10623 Berlin, Germany, DE
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Abstract: | Summary.
This work is concerned with the approximation and
the numerical
computation of polygonal minimal surfaces in
.
Polygonal minimal surfaces correspond to the
critical points of
Shiffman's function
. Since this function is analytic,
polygonal minimal surfaces can be characterized by
means of the second
derivative of .
We present a finite element
approximation of
quasiminimal surfaces together with an error
estimate. In this way we
obtain discrete approximations
of
and of
. In particular we prove that the
discrete functions
converge uniformly on certain compact subsets. This
will be the main
tool for proving existence and convergence of
discrete minimal
surfaces in neighbourhoods of non-degenerate
minimal surfaces. In the
numerical part of this paper we compute numerical
approximations of
polygonal minimal surfaces by use of Newton's
method applied to .
Received October 27, 1994 |
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Keywords: | |
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