Finite-element approximation of 2D elliptic optimal design

We consider a problem of elliptic optimal design in two space dimensions. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n = 2, Sverak [V. Sverak, On optimal shape design, J. Math. Pures Appl. 72 (1993) 537-551] proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complementary sets have a uniformly bounded number of connected components. The proof in [V. Sverak, On optimal shape design, J. Math. Pures Appl. 72 (1993) 537-551] is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology H-c and the continuous dependence of the solutions of the Dirichlet Laplacian in H-1 with respect to it. In this article we introduce a finite-element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. The discrete optimal design problem admits at least one solution since it is a finite optimization problem. We prove that any limit in H-c of discrete optimal shapes, when the mesh-size tends to zero, is an optimal domain for the continuous optimal design problem. The proof relies on the following two key facts: (a) any open bounded set of R-2 can be approximated in H-c by a sequence of triangulated domains, (b) finite-element approximations of the Dirichlet Laplacian in the triangulated domains converge in H-1 to the solutions of the continuous Dirichlet problem whenever the triangulated domains converge in H-c. (c) 2005 Elsevier SAS. All rights reserved.