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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

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