Previous |  Up |  Next


dual finite element method; optimal domain; Thomson principle; rate of convergence; numerical examples
An optimal part of the boundary of a plane domain for the Poisson equation with mixed boundary conditions is to be found. The cost functional is (i) the internal energy, (ii) the norm of the external flux through the unknown boundary. For the numerical solution of the state problem a dual variational formulation - in terms of the gradient of the solution - and spaces of divergence-free piecewise linear finite elements are used. The existence of an optimal domain and some convergence results are proved.
[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. DOI 10.1007/BF01447854 | MR 0443372
[2] J. Haslinger I. Hlaváček: Convergence of a finite element method based on the dual variational formulation. Apl. Mat. 21 (1976), 43 - 65. MR 0398126
[3] I. Hlaváček: The density of solenoidal functions and the convergence of a dual finite element method. Apl. Mat. 25 (1980), 39-55. MR 0554090
[4] I. Hlaváček: Dual finite element analysis for some elliptic variational equations and inequalities. Acta Applic. Math. 1, (1983), 121 - 20. DOI 10.1007/BF00046832 | MR 0713475
[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.
[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. MR 0684830
[7] M. Křížek: Conforming equilibrium finite element methods for some elliptic plane problems. R.A.I.R.O. Anal. numér, 17, (1983), 35-65. MR 0695451
[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. MR 0845923
[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. DOI 10.1007/BF01588950 | MR 0462607 | Zbl 0383.90074
[11] R. Fletcher: Practical methods of optimization, vol. 2, constrained optimization. J. Wiley, Chichester, 1981. MR 0633058 | Zbl 0474.65043
Partner of
EuDML logo