Abstract
In this paper we consider a problem of parabolic optimal design in 2D for the heat equation with Dirichlet boundary conditions. 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. We prove that, as the mesh size tends to zero, any limit in H of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. Sverak in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.
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