Abstract
In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain $\Omega$. To solve this problem numerically, it is usually necessary to approximate $\Omega$ by a (typically polygonal) new domain $\Omega_h$. The difference between the solutions of both infinite-dimensional control problems, one formulated in $\Omega$ and the second in $\Omega_h$, was studied in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746–3780], where an error of order $O(h)$ was proved. In [K. Deckelnick, A. Günther, and M. Hinze, SIAM J. Control Optim., 48 (2009), pp. 2798–2819], the numerical approximation of the problem defined in $\Omega$ was considered. The authors used a finite element method such that $\Omega_h$ was the polygon formed by the union of all triangles of the mesh of parameter h. They proved an error of order $O(h^{3/2})$ for the difference between continuous and discrete optimal controls. Here we show that the estimate obtained in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746–3780] cannot be improved, which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from $\Omega$ to $\Omega_h$.
Highlights
In this paper we are concerned with the approximation of the control problem⎧ ⎪⎪⎪⎨ min J(u) = N L(x, yu(x)) dx + Ω u2(x) dσ(x) Γ (P)⎪⎪⎪⎩ subject to ∈ (L∞(Ω) ∩ H1/2(Ω)) × L∞(Γ), α ≤ u(x) ≤ β for a.e. x ∈ Γ, where the state yu associated to the control u is the solution of the Dirichlet problem (1.1)−Δy + a(x, y) = 0 in Ω, y = u on Γ.Ω is an open, convex, and bounded subset of R2 with a C2 boundary Γ
We study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain Ω
1 2 (y yd(x))2 in the cost functional. Their goal was different: they discretized the control problem by using finite elements associated to a triangulation of the polygonal domain Ωh
Summary
⎪⎪⎪⎩ subject to (yu, u) ∈ (L∞(Ω) ∩ H1/2(Ω)) × L∞(Γ), α ≤ u(x) ≤ β for a.e. x ∈ Γ, where the state yu associated to the control u is the solution of the Dirichlet problem (1.1). Their goal was different: they discretized the control problem by using finite elements associated to a triangulation of the polygonal domain Ωh. The previous example is inspired in another one given by Thomee [8] to prove that the estimates derived by him in the approximation of Dirichlet’s problem were sharp. He considered the adjoint state equation (2.3) as the example of Dirichlet’s problem. Let uand uh denote the solutions of problems (P) and (Ph); there exists a constant C > 0, independent of h, such that the following estimate holds:.
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