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$L^2$-estimate for the discrete Plateau Problem

Author(s): Paola Pozzi.
Journal: Math. Comp. 73 (2004), 1763-1777.
MSC (2000): Primary 65N30; Secondary 49Q05, 53A10
Posted: December 22, 2003
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Abstract: In this paper we prove the $L^2$ convergence rates for a fully discrete finite element procedure for approximating minimal, possibly unstable, surfaces.

Originally this problem was studied by G. Dziuk and J. Hutchinson. First they provided convergence rates in the $H^1$ and $L^2$ norms for the boundary integral method. Subsequently they obtained the $H^1$ convergence estimates using a fully discrete finite element method. We use the latter framework for our investigation.

References:

1.
G. Dziuk and J. E. Hutchinson, On the Approximation of Unstable Parametric Minimal Surfaces, J. Calc. Var. 4 (1996), 27-58. MR 96m:49073

2.
G. Dziuk and J. E. Hutchinson, $L^2$ Estimates for Approximations to Minimal Surfaces, Proceedings of the International Conference: Curvature Flows and Related Topics, 67-92, A. Damlamian, J. Spruck and A. Visintin, eds., Gakkotosho, Tokyo, 1995. MR 97g:58042 MR 2000a:65144

3.
G. Dziuk and J. E. Hutchinson, The Discrete Plateau Problem: Algorithm and Numerics, Math. Comp. 68 (1999), no. 225, 1-23. MR 2000a:65144

4.
G. Dziuk and J. E. Hutchinson, The Discrete Plateau Problem: Convergence Results, Math. Comp. 68 (1999), no. 226, 519-546. MR 2000a:65145

5.
G. Dziuk and J. E. Hutchinson, Finite Element Approximations to Surfaces of Prescribed Variable Mean Curvature, preprint.

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Additional Information:

Paola Pozzi
Affiliation: Centre for Mathematics and its Applications, MSI, Australian National University, Canberra, Australian Capital Territory 0200, Australia
Email: Paola.Pozzi@maths.anu.edu.au

DOI: 10.1090/S0025-5718-03-01630-2
PII: S 0025-5718(03)01630-2
Keywords: Minimal surfaces, finite elements, order of convergence, Plateau Problem
Received by editor(s): June 12, 2002
Received by editor(s) in revised form: March 11, 2003
Posted: December 22, 2003
Copyright of article: Copyright 2003, American Mathematical Society

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