The discrete plateau problem:

Convergence results

Authors:
Gerhard Dziuk and John E. Hutchinson

Journal:
Math. Comp. **68** (1999), 519-546

MSC (1991):
Primary 65N30; Secondary 49Q05, 53A10

DOI:
https://doi.org/10.1090/S0025-5718-99-01026-1

MathSciNet review:
1613699

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Abstract | References | Similar Articles | Additional Information

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.

**[Be]**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****[LM]**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:
https://doi.org/10.1090/S0025-5718-99-01026-1

Keywords:
Minimal surface,
finite elements,
order of convergence,
Plateau Problem

Received by editor(s):
August 26, 1996

Article copyright:
© Copyright 1999
American Mathematical Society