Shape optimization by means of the penalty method with extrapolation. (English) Zbl 0826.65056

Model state problems described by a Poisson equation in a bounded two- dimensional domain with homogeneous Dirichlet boundary condition are considered. The optimal shape design problem is defined by two frequently used cost functionals.
Error estimates are derived. It is shown that the rate of convergence of the penalty method can be increased by means of an extrapolation. The existence of a solution to the approximate optimization problem is discussed and a convergence theorem is proved.
Reviewer: R.Tracht (Essen)


65K10 Numerical optimization and variational techniques
49M30 Other numerical methods in calculus of variations (MSC2010)
49J20 Existence theories for optimal control problems involving partial differential equations
Full Text: DOI EuDML


[1] I. Babuška: The finite element method with penalty. Math. Comp. 27 (1973), 221-228. · Zbl 0299.65057 · doi:10.2307/2005611
[2] I. Babuška: Numerical solution of partial differential equations. Preprint, March 1973, Univ. of Maryland.
[3] P.G. Ciarlet: Basic error estimates for elliptic problems. Handbook of Numer. Anal., vol. II, Finite element methods (Part 1) by P.G. Ciarlet and J.L. Lions, Elsevier, (North-Holland), 1991. · Zbl 0875.65086
[4] S. Conte, C. de Boor: Elementary numerical analysis; an algorithmic approach. McGraw-Hill, New York, 1972. · Zbl 0257.65002
[5] P. Grisvard: Boundary value problems in non-smooth domains. Univ. of Maryland, Lecture Notes #19, 1980.
[6] P. Grisvard: Singularities in boundary value problems. RMA 22, Res. Notes in Appl. Math., Masson, Paris, Springer-Verlag, Berlin, 1992. · Zbl 0778.93007
[7] E.J. Haug, K.K. Choi, V. Komkov: Design sensitivity analysis of structural systems. Academic Press, Orlando-London, 1986. · Zbl 0618.73106
[8] I. Hlaváček: Penalty method and extrapolation for axisymmetric elliptic problems with Dirichlet boundary conditions. Apl. Mat. 35 (1990), 405-417. · Zbl 0725.65098
[9] J. Chleboun, R. Mäkinen: Primal formulation of an elliptic equation in smooth optimal shape problems. Advances in Math. Sci. Appl.
[10] J. Kadlec: On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of a convex open set. Czechoslovak Math. J. 14 (1964), no. 89, 386-393.. · Zbl 0166.37703
[11] J.T. King: New error bounds for the penalty method and extrapolation. Numer. Math. 23 (1974), 153-165. · Zbl 0272.65092 · doi:10.1007/BF01459948
[12] J.T. King, S.M. Serbin: Boundary flux estimates for elliptic problems by the perturbed variational method. Computing, 16 (1976), 339-347. · Zbl 0338.65054 · doi:10.1007/BF02252082
[13] J.T. King, S.M. Serbin: Computational experiments and techniques for the penalty method with extrapolation. Math. Comp. 32 (1978), 111-126. · Zbl 0384.65057 · doi:10.2307/2006260
[14] J.L. Lions, E. Magenes: Problèmes aux limites non homogènes et applications. vol. 1, Dunod, Paris, 1968. · Zbl 0212.43801
[15] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. · Zbl 1225.35003
[16] M. Zlámal: Curved elements in the finite element method I. SIAM J. Num. Anal. 10 (1973), 229-240. · Zbl 0285.65067 · doi:10.1137/0710022
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