A priori error estimates for finite element discretizations of a shape optimization problem
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1733-1763.

In this paper we consider a model shape optimization problem. The state variable solves an elliptic equation on a domain with one part of the boundary described as the graph of a control function. We prove higher regularity of the control and develop a priori error analysis for the finite element discretization of the shape optimization problem under consideration. The derived a priori error estimates are illustrated on two numerical examples.

DOI : 10.1051/m2an/2013086
Classification : 49Q10, 49M25, 65M15, 65M60
Mots clés : shape optimization, existence and convergence of approximate solutions, error estimates, finite elements
@article{M2AN_2013__47_6_1733_0,
     author = {Kiniger, Bernhard and Vexler, Boris},
     title = {\protect\emph{A priori }error estimates for finite element discretizations of a shape optimization problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1733--1763},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {6},
     year = {2013},
     doi = {10.1051/m2an/2013086},
     zbl = {1283.49051},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2013086/}
}
TY  - JOUR
AU  - Kiniger, Bernhard
AU  - Vexler, Boris
TI  - A priori error estimates for finite element discretizations of a shape optimization problem
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 1733
EP  - 1763
VL  - 47
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2013086/
DO  - 10.1051/m2an/2013086
LA  - en
ID  - M2AN_2013__47_6_1733_0
ER  - 
%0 Journal Article
%A Kiniger, Bernhard
%A Vexler, Boris
%T A priori error estimates for finite element discretizations of a shape optimization problem
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 1733-1763
%V 47
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2013086/
%R 10.1051/m2an/2013086
%G en
%F M2AN_2013__47_6_1733_0
Kiniger, Bernhard; Vexler, Boris. A priori error estimates for finite element discretizations of a shape optimization problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1733-1763. doi : 10.1051/m2an/2013086. http://archive.numdam.org/articles/10.1051/m2an/2013086/

[1] Gascoigne: The finite element toolkit. http://www.gascoigne.uni-hd.de/

[2] Rodobo: A c++ library for optimization with stationary and nonstationary pdes. http://rodobo.uni-hd.de/

[3] Y.A. Alkhutov and V.A. Kondratev, Solvability of the Dirichlet problem for second-order elliptic equations in a convex domain. Differentsial′nye Uravneniya 28 (1992) 806-818, 917. | MR | Zbl

[4] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, vol. 34, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1993). | MR | Zbl

[5] D. Braess, Finite Elemente, Springer-Verlag (2007). | Zbl

[6] E. Casas and F. Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybernet. 31 (2002) 695-712. | MR | Zbl

[7] E. Casas and F. Tröltzsch, A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput. Optim. Appl. 53 (2012) 173-206. | MR | Zbl

[8] D. Chenais and E. Zuazua, Controllability of an elliptic equation and its finite difference approximation by the shape of the domain. Numer. Math. 95 (2003) 63-99. | MR | Zbl

[9] D. Chenais and E. Zuazua, Finite-element approximation of 2D elliptic optimal design. J. Math. Pures Appl. 85 (2006) 225-249. | MR | Zbl

[10] T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441-463. | MR | Zbl

[11] K. Eppler, H. Harbrecht, and R. Schneider, On convergence in elliptic shape optimization. SIAM J. Control Optim. 46 (2007) 61-83 (electronic). | MR

[12] P. Grisvard, Elliptic problems in nonsmooth domains, vol. 24, Monographs and Studies in Mathematics, Pitman. Advanced Publishing Program, Boston, MA (1985). | MR | Zbl

[13] J. Haslinger and R.A.E. Mäkinen, Introduction to shape optimization. Theory, approximation, and computation, vol. 7, Advances in Design and Control, Society for Industrial and Applied Mathematics SIAM. Philadelphia, PA (2003). | MR | Zbl

[14] J. Haslinger and P. Neittaanmäki, Finite element approximation for optimal shape, material and topology design. John Wiley & Sons Ltd., Chichester, 2nd edition (1996). | MR | Zbl

[15] K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications, vol. 15, Advances in Design and Control, Society for Industrial and Applied Mathematics. SIAM, Philadelphia, PA (2008). | MR | Zbl

[16] D.S. Jerison and C.E. Kenig, The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. (N.S.) 4 (1981) 203-207. | MR | Zbl

[17] D.S. Jerison and C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161-219. | MR | Zbl

[18] J. Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain. Czechoslovak Math. J. 14 (1964) 386-393. | MR | Zbl

[19] K. Kunisch and G. Peichl, Numerical gradients for shape optimization based on embedding domain techniques. Comput. Optim. Appl. 18 (2001) 95-114. | MR | Zbl

[20] M. Laumen, A comparison of numerical methods for optimal shape design problems. Optim. Methods Softw. 10 (1999) 497-537. | MR | Zbl

[21] M. Laumen, Newton's method for a class of optimal shape design problems. SIAM J. Optim. 10 (2000) 503-533 (electronic). | Zbl

[22] J. Nečas, Sur la coercivité des formes sesquilinéaires, elliptiques. Rev. Roumaine Math. Pures Appl. 9 (1964) 47-69. | MR | Zbl

[23] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38 (1982) 437-445. | MR | Zbl

[24] G. Savaré, Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998) 176-201. | MR | Zbl

[25] T. Slawig, Shape optimization for semi-linear elliptic equations based on an embedding domain method. Appl. Math. Optim. 49 (2004) 183-199. | MR | Zbl

[26] J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization, Shape sensitivity analysis, vol. 16, Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1992). | Zbl

[27] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen, Vieweg+Teubner (2009).

Cité par Sources :