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The discrete plateau problem: Convergence results
Author(s):
Gerhard
Dziuk;
John
E.
Hutchinson.
Journal:
Math. Comp.
68
(1999),
519-546.
MSC (1991):
Primary 65N30;
Secondary 49Q05, 53A10
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Abstract:
We solve the problem of finding and justifying an optimal fully discrete finite element procedure for approximating minimal, including unstable, surfaces. In a previous paper we introduced the general framework and some preliminary estimates, developed the algorithm and give the numerical results. In this paper we prove the convergence estimate.
References:
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- M. Berger, Nonlinearity and Functional Analysis, Academic Press 1977. MR 58:7671
- [BPS]
- J.H. Bramble, J.E. Pasciak & A.H. Schatz, An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures, Math. Comp. 46 (1986), 361-369. MR 88a:65123
- [DH0]
- G. Dziuk, J.E. Hutchinson, On the approximation of unstable parametric minimal surfaces, preprint No. 340 (1994) SFB 256, Bonn, or CMA Math. Res. Rep. 9 (1994), Australian National University.
- [DH1]
- G. Dziuk, J.E. Hutchinson, On the approximation of unstable parametric minimal surfaces, Calc. Var. 4 (1996), 27-58. MR 96m:49073
- [DH3]
- G. Dziuk, J.E. Hutchinson, A Finite Element Method for the Computation of Parametric Minimal Surfaces, Tatra Mount. Math. J. 4 (1994), 49-62. MR 95g:65151
- [DH4]
- G. Dziuk, J.E. Hutchinson, The Discrete Plateau Problem: Algorithm and Numerics, Math. Comp. 68 (1999), 1-23.
- [JK]
- D.S. Jerison & C.E. Kenig, Boundary Value Problems on Lipschitz Domains, Studies in Mathematics 23, 1-68, ed. W. Littman, MAA Studies in Mathematics. MR 85f:35057
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- J.L. Lions & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Grundlehren der Mathematischen Wissenschaften 181, Springer-Verlag 1972. MR 50:2670
- [St]
- M. Struwe, Plateau's Problem and the Calculus of Variations, Mathematical Notes 35, Princeton University Press 1988. MR 90h:58016
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Additional Information:
Gerhard
Dziuk
Affiliation:
Institut für Angewandte Mathematik, Universität Freiburg, Hermann--Herder--Str. 10, D-79104 Freiburg i. Br., GERMANY
Email:
gerd@mathematik.uni-freiburg.de
John
E.
Hutchinson
Affiliation:
Department of Mathematics, School of Mathematical Sciences, Australian National University, GPO Box 4, Canberra, ACT 0200, AUSTRALIA
Email:
John.Hutchinson@anu.edu.au
DOI:
10.1090/S0025-5718-99-01026-1
PII:
S 0025-5718(99)01026-1
Keywords:
Minimal surface,
finite elements,
order of convergence,
Plateau Problem
Received by editor(s):
August 26, 1996
Copyright of article:
Copyright
1999,
American Mathematical Society
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