A Paradox in the Approximation of Dirichlet Control Problems in Curved Domains

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Ω is an open, convex, and bounded subset of R 2 with a C 2 boundary Γ.The boundedness of the control is required to deal with the nonlinearity of the state equation * Received by the editors May 11, 2010; accepted for publication (in revised form) July 20, 2011; published electronically September 27, 2011.The first and the third authors were partially supported by the Spanish Ministry of Science and Innovation under projects MTM2008-04206 and "Ingenio Mathematica (i-MATH)" CSD2006-00032 (Consolider Ingenio 2010).http://www.siam.org/journals/sicon/49-5/79488.html† Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I.Industriales y de Telecomunicación, Universidad de Cantabria, Av.Los Castros s/n, 39005 Santander, Spain (eduardo.casas@unican.es).
and the integrand L(x, y u ), but it is not necessary for linear equations and functionals L with a quadratic polynomial growth in y.
To solve this problem it is necessary to approximate Ω by a simpler domain (typically polygonal) Ω h with a boundary Γ h .In a recent paper Casas and Sokolowski [4] studied the influence of the replacement of Ω by Ω h on the solutions of the control problems.To this aim, a polygonal approximation Ω h of Ω was considered, h being the maximum length of the sides of the polygon.Then a one-to-one mapping g h : Γ h −→ Γ was defined and a control problem (P h ) was formulated in Ω h in a similar way to (P).The convergence of the these approximations was proved in the following sense: 1.For any sequence {ū h } h>0 of solutions of control problems (P h ), the sequence (Γ) and any weak limit ū is a solution of (P). 2. For any strict local minimum of (P), ū, there exists a sequence {ū h } h>0 such that ūh is a local solution of (P h ) and ≤ Ch was proved for local solutions of (P) and (P h ) such that ūh • g −1 h ū weakly in H 1/2 (Γ).On the other hand, Deckelnick, Günther, and Hinze [5] studied the problem (P) without the nonlinear term in the state equation, a(x, y) = −f (x), with f ∈ L 2 (Ω), and taking L(x, y) = 1 2 (y − y d (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 Ω h .In this case the control problems (P) and (P h ) have a unique solution ū and ūh , respectively.Under a nonrestrictive assumption in practice on the triangulation of Ω h , they proved the error estimate ū − ūh If we compare the results of [4] and [5], the difference is surprising.In [4] the problem (P h ) is the same as the problem (P) except for the change of domain Ω by Ω h , but there is no discretization of the control problem.In [5] the control problem (P h ) is a discrete problem where Ω has been replaced by Ω h and the partial differential equation has been discretized so that the states y u are approximated by piecewise linear functions y h (u) solving the discrete variational equation.However, in the second case we get a better approximation to ū than in the first case.The reader could conclude that the error estimates of [4] are not sharp and should be improved.In this paper we provide an example showing that the error estimates of [4] cannot be improved.Nevertheless, as predicted by the theory, the numerical computation on this example confirms the order h 3/2 for the difference among ū and the solutions of the discrete problems.The goal of this paper is to show this paradox that reminds us of Babuska's paradox.Indeed, Babuska's paradox concerns the approximation of a simply supported circular plate, uniformly loaded, by a sequence of regular polygonal plates inscribed in the circle, also simply supported and uniformly loaded.It happens that the solutions for the polygons do not converge to that of the circle; see [1].In the case we are considering in this paper the convergence holds, but it is not so good as the numerical approximation, which is also rather paradoxical.
The plan of the paper is as follows.In the next section we formulate an example of a control problem falling into the framework defined above, and we prove that the estimates obtained in [4] are optimal for this problem.In section 3 we describe the finite element approximation of the example and show the computational results, which confirm the theoretical estimates proved in [5].Finally, in section 4 we explain why the numerical approach provides a better approximation of ū than the exact solution of (P h ).Downloaded 11/23/22 to 193.144.185.28 .Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy

The example.
In what follows Ω will denote the unit ball of R 2 centered at 0; then Γ is the unit circumference.In this domain we consider the control problem where the state y u associated to the control u is the solution of the Dirichlet problem (2.1) It is obvious that (P) is strictly convex and has a unique solution ū that can be characterized by the optimality system where ν(x) denotes the unit outward normal vector to Γ at the point x.For the selected domain we have that ν(x) = x.It is very easy to check that the solution of the above system is given by Now we define the polygonal domain Ω h .For every positive integer n we consider the points of Γ, For convenience, we set x n+1 = x 1 .It is easy to check that the distance between two consecutive points is h = |x j+1 − x j | = 2 sin π n .We take Γ h as the polygonal line joining the knots {x j } n j=1 , and Ω h is the open domain enclosed by Γ h .In this domain we consider the control problem where the state y h,u h associated to the control u h is the solution of the Dirichlet problem The previous example is inspired in another one given by Thomée [8] to prove that the estimates derived by him in the approximation of Dirichlet's problem were sharp.In fact, he considered the adjoint state equation (2.3) where ν h (x) is the unit outward normal vector to Γ h at the point x; hence if x ∈ (x j , x j+1 ), then ν h (x) = xj+xj+1 |xj+xj+1| .The solution φh of (2.8) is of class C 1 in Ωh (see section 4 for more details), and therefore To compare the solutions ū and ūh we introduce the mapping g h : Γ h −→ Γ as follows.For every 1 ≤ j ≤ n, x j x j+1 denotes the arc of Γ delimited by the points x j and x j+1 .Then we have that Γ = ∪ n j=1 x j x j+1 and Γ h = ∪ n j=1 [x j , x j+1 ].Now we introduce a parametrization of x j x j+1 , where τ j = (x j+1 − x j )/h, ν j is the restriction of ν h to the side (x j , x j+1 ) of Γ h , and Clearly g h is one-to-one.We denote by τ (x) the unit tangent vector to Γ at the point x such that {τ (x), ν(x)} is a direct reference system in R 2 .We can obtain the expressions for these vectors from the given parametrization.If x is a point of the arc x j x j+1 , then where x = x j + tτ j + φ(t)ν j .From these expressions and the properties of φ we deduce that We also have The following result is an immediate consequence of [4, Theorem 9.1] Theorem 2.1.Let ū and ūh denote the solutions of problems (P) and (P h ); then there exists a constant C > 0, independent of h, such that the following estimate holds: Now we prove that this estimate cannot be improved.To get an underestimate for ū − ūh • g −1 h we use (2.4) and (2.9); then

15)
Using that φ = 0 on Γ and (2.10) we get that On the other hand, from the definition of g h and the properties of φ we get From (2.15), (2.16), and (2.17) we conclude Then it is enough to prove the existence of a constant C > 0, independent of h, such that Following Kenig [7, p. 121], φ h ∈ H 3/2 (Ω h ) and the following estimates hold: (2.20) From these inequalities we deduce that Hence, if we prove that φ H 1 (Γ h ) ≥ Ch, then (2.18) is concluded.We have that Using that the angle between x j and x j+1 is 2π/n, we get Therefore,

Now we observe that
Thus, we conclude  Since there are no control constraints, this approach is equivalent to the one given in [5], where the control is not discretized, but it is finally obtained as the pointwise projection of the discrete normal derivative.We will take as mesh size h the length of one side of Γ h .For quasi-uniform meshes this is equivalent to the usual choice of the maximum edge size of the triangulation.With these settings, Theorem 5.4 in [5] states that there exists a constant C > 0 such that Numerical testing confirms this order of convergence.For p = 2 or p = ∞, the experimental error is given by and the experimental order of convergence is .
We obtain the results summarized in Table 3.1.
A picture of the solution for meshes obtained after successive refinements from an octagon is shown in Figure 3.2.Notice that the numerical solution has some needles located at the vertexes of the initial rough mesh.Nevertheless, these deviations are small and the convergence order in L ∞ (Γ h ) is linear on this example.

Explaining the paradox.
The reason for lower accuracy than expected in approximating ū by ūh is found at the vertices x j of the polygonal boundary Γ h .Indeed, from (2.8) we deduce that φh ∈ W 2,p h (Ω h ) for some p h > 2 depending on Downloaded 11/23/22 to 193.144.185.28 .Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy the angles of the polygonal domain Ω h ; see Grisvard [6].In fact, it holds for any p h < 2 + 4/(n − 4), where n is the number of vertices of Ω h .The point is that W 2,p h (Ω h ) ⊂ C 1 ( Ωh ).Then, taking into account that φh = 0 on Γ h , we have at every vertex x j ∇ φh (x j ) • τ j−1 = ∇ φh (x j ) • τ j = 0, and therefore ∇ φh (x j ) = 0. Hence, ∇ φh • ν h is a continuous function on Γ h if we take the value zero on the vertices x j .Even more, we have that Thus, the singularities of Γ h on the vertices x j force the optimal controls ūh to vanish on them.Taking into account that ū(x j ) = −2, we observe a big error between ū and ūh at the vertices.Notice that the number of vertices tends to infinity when h → 0 and {x j } n j=1 becomes dense in Γ.This does not happen if we consider the numerical approximation of ū on Γ h .Indeed, the discrete optimal control is given by (3.1), where we use the discrete normal derivative of the discrete adjoint state; see (3.2).This discrete normal derivative does not vanish necessarily at the vertices x j .
Let us finish by showing the solution ūh of (P h ).To compute ūh we make a finite element approximation of (P h ).For that purpose we take a quasi-uniform family of triangulations T h ρ of Ω h (see Figure 4.1).Associated to T h ρ we consider the spaces , and U h ρ the restriction to Γ h of functions in X h ρ .We consider the following approximation of (P h ): where the discrete state y h ρ (u h ρ ) ∈ X h ρ associated to the control u h ρ is the unique solution of the following finite-dimensional problem: