Optimization of the domain in elliptic problems by the dual finite element method. (English) Zbl 0575.65103

Two optimization problems for a class of domains in \({\mathbb{R}}^ 2\) are studied. Each domain is a subset of a fixed rectangle; its boundary has a curved part, the remainder coincides with the boundary of the rectangle. Let y be the solution of the state problem for such a domain \((-\Delta y=f\), mixed homogeneous boundary conditions, Dirichlet condition on the curved part). A domain is said to be optimal if (i) the Dirichlet integral of y in the domain or (ii) the norm of the outward flux through the boundary of the domain is minimal. For both problems the existence of an optimal domain is shown. For the numerical solution a dual variational formulation of the state problem (Thomson principle) is used; finite element subspaces of divergence-free piecewise linear functions are employed. An analysis of convergence is presented. The rate of convergence as well as numerical examples are not given.
Reviewer: J.Weisel


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J20 Variational methods for second-order elliptic equations
Full Text: EuDML


[1] D. Begis R. Glowinski: Application de la méthode des éléments finis à l’approximation d’un problème de domaine optimal. Appl. Math. & Optim. 2 (1975), 130-169. · Zbl 0323.90063
[2] J. Haslinger I. Hlaváček: Convergence of a finite element method based on the dual variational formulation. Apl. Mat. 21 (1976), 43 - 65.
[3] I. Hlaváček: The density of solenoidal functions and the convergence of a dual finite element method. Apl. Mat. 25 (1980), 39-55.
[4] I. Hlaváček: Dual finite element analysis for some elliptic variational equations and inequalities. Acta Applic. Math. 1, (1983), 121 - 20.
[5] J. Haslinger J. Lovíšek: The approximation of the optimal shape problem governed by a variational inequality with flux cost functional. To appear in Proc. Roy. Soc. Edinburgh. · Zbl 0625.73025
[6] I. Hlaváček J. Nečas: Optimization of the domain in elliptic unilateral boundary value problems by finite element method. R.A.I.R.O. Anal. numér, 16, (1982), 351 - 373. · Zbl 0496.65057
[7] M. Křížek: Conforming equilibrium finite element methods for some elliptic plane problems. R.A.I.R.O. Anal. numér, 17, (1983), 35-65. · Zbl 0541.76003
[8] J. Haslinger P. Neittaanmäki: On optimal shape design of systems governed by mixed Dirichlet-Signorini boundary value problems. To appear in Math. Meth. Appl. Sci. · Zbl 0603.49020
[9] P. Neittaanmäki T. Tiihonen: Optimal shape design of systems governed by a unilateral boundary value problem. Lappeenranta Univ. of Tech., Dept. of Physics and Math., Res. Kept. 4/1982.
[10] B. A. Murtagh: Large-scale linearly constrained optimization. Math. Programming, 14 (1978), 41-72. · Zbl 0383.90074
[11] R. Fletcher: Practical methods of optimization, vol. 2, constrained optimization. J. Wiley, Chichester, 1981. · Zbl 0474.65043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.