# A priori error estimates for finite element discretizations of a shape optimization problem

Bernhard Kiniger; Boris Vexler

- Volume: 47, Issue: 6, page 1733-1763
- ISSN: 0764-583X

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topKiniger, Bernhard, and Vexler, Boris. "A priori error estimates for finite element discretizations of a shape optimization problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.6 (2013): 1733-1763. <http://eudml.org/doc/273318>.

@article{Kiniger2013,

abstract = {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.},

author = {Kiniger, Bernhard, Vexler, Boris},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {shape optimization; existence and convergence of approximate solutions; error estimates; finite elements; approximate solutions},

language = {eng},

number = {6},

pages = {1733-1763},

publisher = {EDP-Sciences},

title = {A priori error estimates for finite element discretizations of a shape optimization problem},

url = {http://eudml.org/doc/273318},

volume = {47},

year = {2013},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 6

SP - 1733

EP - 1763

AB - 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.

LA - eng

KW - shape optimization; existence and convergence of approximate solutions; error estimates; finite elements; approximate solutions

UR - http://eudml.org/doc/273318

ER -

## References

top- [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. Zbl0834.35038MR1198129
- [4] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, vol. 34, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1993). Zbl0818.47059MR1225101
- [5] D. Braess, Finite Elemente, Springer-Verlag (2007). Zbl0754.65084
- [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. Zbl1126.49315MR1978747
- [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. Zbl1264.49030MR2964840
- [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. Zbl1045.93023MR1993939
- [9] D. Chenais and E. Zuazua, Finite-element approximation of 2D elliptic optimal design. J. Math. Pures Appl.85 (2006) 225–249. Zbl1086.49027MR2199013
- [10] T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput.34 (1980) 441–463. Zbl0423.65009MR559195
- [11] K. Eppler, H. Harbrecht, and R. Schneider, On convergence in elliptic shape optimization. SIAM J. Control Optim. 46 (2007) 61–83 (electronic). Zbl05240370MR2299620
- [12] P. Grisvard, Elliptic problems in nonsmooth domains, vol. 24, Monographs and Studies in Mathematics, Pitman. Advanced Publishing Program, Boston, MA (1985). Zbl0695.35060MR775683
- [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). Zbl1020.74001MR1969772
- [14] J. Haslinger and P. Neittaanmäki, Finite element approximation for optimal shape, material and topology design. John Wiley & Sons Ltd., Chichester, 2nd edition (1996). Zbl0845.73001MR1419500
- [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). Zbl1156.49002MR2441683
- [16] D.S. Jerison and C.E. Kenig, The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. (N.S.) 4 (1981) 203–207. Zbl0471.35026MR598688
- [17] D.S. Jerison and C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal.130 (1995) 161–219. Zbl0832.35034MR1331981
- [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. Zbl0166.37703MR170088
- [19] K. Kunisch and G. Peichl, Numerical gradients for shape optimization based on embedding domain techniques. Comput. Optim. Appl.18 (2001) 95–114. Zbl0970.90114MR1818916
- [20] M. Laumen, A comparison of numerical methods for optimal shape design problems. Optim. Methods Softw.10 (1999) 497–537. Zbl0933.49028MR1688679
- [21] M. Laumen, Newton’s method for a class of optimal shape design problems. SIAM J. Optim. 10 (2000) 503–533 (electronic). Zbl0956.65053
- [22] J. Nečas, Sur la coercivité des formes sesquilinéaires, elliptiques. Rev. Roumaine Math. Pures Appl.9 (1964) 47–69. Zbl0196.40701MR179457
- [23] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comput.38 (1982) 437–445. Zbl0483.65007MR645661
- [24] G. Savaré, Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal.152 (1998) 176–201. Zbl0889.35018MR1600081
- [25] T. Slawig, Shape optimization for semi-linear elliptic equations based on an embedding domain method. Appl. Math. Optim.49 (2004) 183–199. Zbl1077.49031MR2033834
- [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). Zbl0761.73003
- [27] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen, Vieweg+Teubner (2009).

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