A posteriori error analysis for a distributed optimal control problem governed by the von K\'{a}rm\'{a}n equations

This article discusses numerical analysis of the distributed optimal control problem governed by the von K\'{a}rm\'{a}n equations defined on a polygonal domain in $\mathbb{R}^2$. The state and adjoint variables are discretised using the nonconforming Morley finite element method and the control is discretized using piecewise constant functions. A priori and a posteriori error estimates are derived for the state, adjoint and control variables. The a posteriori error estimates are shown to be efficient. Numerical results that confirm the theoretical estimates are presented.


Problem formulation
Let Ω ⊂ R 2 be a polygonal domain and denotes the outward normal vector to the boundary Ω of Ω.This paper considers the distributed control problem governed by the von Kármán equations stated below: min ∈ J (Ψ, ) subject to (1.1a)

Motivation
The optimal control problem governed by the von Kármán equations (1.1a)-(1.1c) is analysed in [31] for 1 conforming finite elements.In [19], a priori error estimates are derived under minimal regularity assumptions on INTRODUCTION 2 the exact solution where the state and adjoint variables are discretised using Morley finite element methods (FEMs).The discrete trilinear form in the weak formulation [19] is derived after an integration by parts.In this article, a simplified form of the trilinear form that involves the von Kármán bracket itself is considered.This choice of the trilinear form [11] is appropriate for both reliable and efficient a posteriori estimates.To the best of our knowledge, there are no results in literature that discuss a posteriori error analysis for the approximation of regular solutions of optimal control problems governed by von Kármán equations.Recently, a posteriori error analysis for the optimal control problem governed by second-order stationary Navier-Stokes equations is studied in [1] with conforming finite element method under smallness assumption on the data.The trilinear form in [1] vanishes whenever the second and third variables are equal, and satisfies the anti-symmetric property with respect to the second and third variables and this aids the a posteriori error analysis.This paper discusses approximation of regular solutions for fourth-order semi-linear problems without any smallness assumption on the data.Moreover, the trilinear form for von Kármán equations does not satisfy the properties stated above and hence leads to additional challenges in the analysis.
The von Kármán equations [21] that describes the bending of very thin elastic plates offers challenges in its numerical approximation; mainly due to its nonlinearity and higher order nature; we refer to [2-5, 21, 27] and the references therein for the existence of solutions, regularity and bifurcation phenomena of the von Kármán equations.The numerical analysis of von Kármán equations has been studied using conforming FEMs in [9,29], nonconforming Morley FEM in [12,30], mixed FEMs in [17,32], discontinuous Galerkin methods and 0 interior penalty methods in [6,11].
Nonconforming Morley FEM based on piecewise quadratic polynomials in a triangle is more elegant, attractive and simpler for fourth-order problems.However, the convergence analysis offers a lot of novel challenges in the context of control problems governed by semilinear problems with trilinear nonlinearity since the discrete space M is not a subspace of 2 0 (Ω).The adjoint variable in the control problem satisfies a fourth-order linear problem with lower-order terms and its a priori and a posteriori analysis with Morley FEM offers additional difficulties.
The regularity results of von Kármán equations in [5] extends to the regularity of the state and adjoint variables of the control problem [31] and ensures that the optimal state and adjoint variables belong to 2 0 (Ω) ∩ 2+ (Ω), where ∈ ( 1  2 , 1], referred to as the index of elliptic regularity, is determined by the interior angles of Ω.Note that when Ω is convex, = 1.

Contributions
In continuous formulation (see (2.1)) and the conforming FEM [31], the trilinear form (•, •, •) is symmetric with respect to all the three variables; that makes the analysis simpler to a certain extent.However, for fourth-order systems, nonconforming Morley FEM is attractive and is a method of choice [12] and this motivated the a priori analysis for the optimal control problem in [19].The expression for the discrete trilinear form NC (•, •, •) in [19] defined as 1   2 ∈T ∫ cof( 2 M ) M • M dx for all Morley functions M , M and M is obtained after an integration by parts, where T denotes the triangulation of Ω.This form is symmetric with respect to the second and third variables.Though this choice of trilinear form leads to optimal order error estimates for the optimal control problem (1.1a)-(1.1c), it leads to terms that involve averages in the reliability analysis of the state equations (as in the case of Navier-Stokes equation considered in [12]).The efficiency estimates are unclear in this context.To overcome this, ] M dx that is symmetric with respect to the first and second variables is chosen in this article.The a priori and a posteriori analysis for the state equations are discussed in [12,13].The a posteriori analysis for the fully discrete optimal control problem governed by von Kármán equations addressed in this article is novel and involves additional difficulties.For instance, the adjoint system in this case involves lower-order terms with leading biharmonic operators.A posteriori analysis for biharmonic operator with lower-order terms is a problem of independent interest.
Thus the contributions of this article can be summarized as follows.
• For a formulation that is different from that in [19], optimal order a priori error estimates in energy norm when state and adjoint variables are approximated by Morley FEM and linear order of convergence for control variable in 2 norm when control is approximated using piece-wise constants are outlined.• Reliable and efficient a posteriori error estimates that drive the adaptive refinement for the optimal state and adjoint variables in the energy norm and control variable in the 2 norm are developed.The approach followed in this paper provides a strategy for the nonconforming FEM analysis of optimal control problems governed by higher-order semi-linear problems.
• Several auxiliary results that are derived will be of interest in other applications -for example, optimal control problems governed by Navier-Stokes problems in the stream-vorticity formulation.• The paper illustrates results of computational experiments that validate both theoretical a priori and a posteriori estimates for the optimal control problem under consideration.

Organisation
The remaining parts of this paper are organised as follows.Section 2 presents the weak and the nonconforming finite element formulations for (1.1a)-(1.1c).The state and adjoint variables are discretised using Morley finite elements and the control variable is discretised using piecewise constant functions.Section 2.3 deals some preliminaries related to Morley FEM.The boundedness properties of the discrete bilinear and trilinear forms that are crucial for the error analysis are discussed in this section.A priori error estimates for the state, adjoint and control variables under minimal regularity assumptions on the exact solution are stated in Section 3. Note that the analysis differs from [19] due to a different trilinear form.Section 4 develops reliable a posteriori estimates for the state, adjoint and control variables of the optimal control problem.Section 5 establishes efficiency results for the optimal control problem.
Results of numerical experiments that validate theoretical estimates are presented in Section 6.Finally, details of proofs of some results stated in Section 3 are derived in the Appendix.(Ω)) is used to denote the product space

Notations
).The notation (resp. ) means there exists a generic mesh independent constant such that ≤ (resp.≥ ).The positive constants appearing in the inequalities denote generic constants which do not depend on the mesh-size.

Weak and Finite Element Formulations
In this section, the weak and Morley FEM formulations for (1.1a)-(1.1c)and some auxiliary results are presented.
For a given ∈ 2 ( ), a solution Ψ of (2.1b)-(2.1c) is said to be regular [19,Definition 2.1] if the linearized form is well-posed.In this case, the pair (Ψ, ) also is referred to as a regular solution to (1.1b)-(1.1c).The pair ( Ψ, ¯ ) ∈ V × is a local solution [16] ) Remark 2.1.The dependence of Ψ with respect to is made explicit with the notation Ψ only when it is necessary.
Remark 2.2.The regular solution Ψ to (2.1) satisfies the inf-sup condition A + B ′ ( Ψ) , Φ , and this leads to The existence of a solution to (2.1) can be obtained using standard arguments of considering a minimizing sequence, which is bounded in V × 2 ( ), and passing to the limit [25,28,33].
The subscripts in the duality pairings are omitted for notational convenience.
The same notation ′ is used either to denote the Fréchet derivative of an operator or the dual of a space, but the context helps to clarify its precise meaning.

Discrete formulation
Let T be an admissible and regular triangulation of the domain Ω into simplices in R 2 , ℎ be the diameter of ∈ T and ℎ := max ∈T ℎ .For a non-negative integer ∈ N 0 , P (T ) denotes the space of piece-wise polynomials of degree at most equal to .Let Π denote the 2 projection onto the space of piece-wise polynomials P (T ).The oscillation of in T reads osc ( , T ) = ℎ 2 ( − Π ) for ∈ N 0 .
The nonconforming Morley element space M is defined by For all M , M and M ∈ M , define the discrete bilinear and trilinear forms by Similarly, for The above definitions of the bilinear and trilinear forms are meaningful for functions in + M (resp.V + V M ).Note that for all , , ∈ , NC ( , ) = ( , ) and NC ( , , ) = ( , , ).
Analogous to the definition of nonlinear operator B : V → V ′ , define the discrete counterparts B NC : (2.7) The admissible space for discrete controls is ℎ, := ∈ 2 ( ) : | ∈ P 0 ( ), ≤ ≤ for all ∈ T .The discrete control problem associated with (2.1) reads min The discrete first order optimality system that comprises of the discrete state and adjoint equations and the first order optimality condition corresponding to (2.8) is where ΘM ∈ V M denotes the discrete adjoint variable that corresponds to the optimal state variable ΨM ∈ V M .

Auxiliary results
This section presents some auxiliary results that are useful to establish the proof of both the a priori as well as a posteriori error estimates.These are very crucial properties of Morley interpolation and companion operators that aids the analysis.Some properties of the discrete bilinear and trilinear forms that will be used throughout in the article are also presented.

Lemma 2.4 (Companion operator
). [15,23] For any M ∈ M , there exists : M → such that Here N ( ) denotes the set of vertices of ∈ T and patch Ω( ) := int ∪ ∈N ( ) ∪ T ( ) , T ( ) denotes the triangles that share the vertex and E (Ω( )) denotes the edges in Ω( ), denotes the unit tangential vector to the edge and [ ] denotes the jump of a function across the edge .
For vector-valued functions, the interpolation and companion operators are to be understood component-wise.

A priori error estimates
This section deals with the a priori error estimates for the state, adjoint and control variables under minimal regularity assumptions on the exact solution.The proof of the results that differ from [19] owing of the choice of the alternate discrete trilinear form are discussed in Appendix.
For a given F, fixed control ∈ and u = ( , 0), consider the auxiliary state equation that seeks Ψ ∈ V such that The nonconforming Morley finite element (FE) approximation to (3.1) seeks The next result on the existence, uniqueness and error estimates of the auxiliary state equation is proved with the help of Lemma A.1 given in the Appendix.The proofs that are available in [13,19] are skipped.Note that a modified proof of Lemma A.1 is presented and it utilises the properties of the companion operator to obtain sharper bounds in comparison to [19,Lemma 3.12].
Remark 3.1.The well-known result for the biharmonic problem for the approximation using Morley nonconforming FEM which states that the 2 error estimate cannot be further improved than that of 1 error estimate [26]  The auxiliary problem corresponding to the adjoint equations seeks Θ ∈ V such that where The existence, uniqueness and convergence results stated in the next theorem follow analogous to that of [19, Theorems 4.1, 4.2 ( )] and is skipped for brevity.
Theorem 3.2 (Existence, uniqueness and energy error estimate).Let ( Ψ, ¯ ) ∈ V × 2 ( ) be a regular solution to (2.1).Then, (i) there exist 0 < 3 ≤ 2 such that, for a sufficiently small choice of discretization parameter and ∈ 3 ( ¯ ), (3.4) admits a unique solution, (ii) for ∈ 3 ( ¯ ) and a sufficiently small choice of discretization parameter, the solutions Θ and Θ ,M of (3.3) and (3.4) satisfy the energy norm error estimate: is the index of the elliptic regularity.
The proof of a priori 1 error estimate stated in the next theorem for adjoint variables is a non-trivial modification of the corresponding result in [19] and is presented in the Appendix.The form of the error estimate will be useful in the adaptive convergence study that is planned for future.

Reliability Analysis
This section deals with the reliability analysis for the a posteriori error estimator for the optimal control problem (2.1).Let T be the set of all admissible triangulations T .Given any 0 < < 1, let T( ) be the set of all triangulations T with mesh-size ≤ for all triangles ∈ T with area | |.Assume that ⊂ Ω is a polygonal domain and that T restricted to yields a triangulation for .The main result of this section is stated first in Theorem 4.1.The proof is presented at the end of this section.Define the auxiliary variable ℎ by where ΘM = ( ¯ M,1 , ¯ M,2 ) is the discrete adjoint variable corresponding to the control ¯ ℎ .

A posteriori error analysis for the adjoint equations
The auxiliary problem that corresponds to the adjoint equations seeks Θ ∈ V such that where ΨM ∈ V M is the solution to (2.9a).Since Ψ is a regular solution to (2.1), the adjoint of the operator in (2.4) satisfies the inf-sup condition given by with the last inequality derived from (2.5b).An introduction of Ψ, the first inequality of (4.16), Lemma 2.7.( ), (4.6) and ≤ /(4 ) show that for any 0 < 2 < , there exists some Φ ∈ V with |||Φ||| 2 = 1 such that with 2 ց 0 in the second last step of the inequality above.This shows the wellposedness of (4.15).A combination of (4.15) and (4.17) leads to a bound for the solution of Θ of (4.15) as where B ′ NC (Ψ) * is the adjoint operator corresponding to B ′ NC (Ψ) and the bounded linear operator (•) (resp.NC (•)) solves the biharmonic system of equations in the sense that for the load ∈ V ′ (resp.∈ V ′ M ), ( , Φ) = , Φ for all Φ ∈ V (resp.NC ( NC , Φ M ) = , Φ M for all Φ M ∈ V M ).A detailed discussion of these operators is provided in Appendix.The next lemma (proved in Appendix) is utilized in the proof of Theorem 4.4.
Proof of Theorem 4.1.The proofs follows from a combination of Theorems 4.2, 4.4 and 4.6 for any T ∈ T( ) which satisfies (4.6).

Efficiency
This section deals with the a posteriori efficient error estimates for the control problem.The local efficiency proofs are based on the standard bubble function techniques [34], [11,Lemma 5.3].The combined result is stated first.
Proof.For each element ∈ T , it can be shown that The proof of (5.3) follows from the standard bubble functions technique.In the proof therein for the first term in the left hand side of (5.3), set : ) 2 in , and zero in Ω \ .The adjoint system (2.5b) with the test function ( , 0), and the symmetric property of ( ) dx = 0.The combination of this, Δ 2 ¯ M,1 = 0 and the arguments in the proof of [11,Lemma 5.3] prove (5.3).The estimates for the second term in the left hand side of (5.3) is analogous to that of the first term.Consider 2 ¯ M,1 (1 − P 0 ) ¯ M,1 2 ( ) , ∈ T .The Hölder's inequality shows that (5.4) The triangle inequality with P 0 M ¯ 1 leads to The inverse inequality [20,Theroem 3.2.6]for the first term, triangle inequality with (1 − P 0 ) ¯ 1 for the second term, projection estimate for P 0 in 2 ( ) [22,Proposition 1.135] and the boundedness property of P 0 prove This with (5.4) result in (5.5) From (4.13), ||| 2 ΨM ||| 2 ( ) ≤ .The estimates for the remaining terms 2 ¯ M,1 (1 − P 0 ) ¯ M,2 2 ( ) , 2 ¯ M,2 (1 − P 0 ) ¯ M,1 2 ( ) follow from similar arguments and hence the details are omitted for brevity.Lemma 2.4.( ) leads to the desired estimate for the edge estimator , ΘM .Analogous terms as the last term in the right hand side of (5.5) are dealt with in [18,Theorem 4.10].
This and Lemma 2.6 result in Here the constant absorbed in ' ′ depends on the shape-regularity of T .This concludes the proof.

Numerical results
The results of the numerical experiments that support the a priori and a posteriori estimates are presented in this section.

Preliminaries
The state and adjoint variables are discretised using the Morley FE and the control variable is discretised using piecewise constant functions.The discrete solution ( ΨM , ΘM , ¯ ℎ ) is computed using a combination of Newtons' method in an inner loop and primal dual active set strategy in an outer loop, see [19,Section 6.1] for the details of the implementation procedure for the a priori case for a different choice of the trilinear form.The initial guess for ( ΨM , ΘM ) in the Newton's iterative scheme is chosen as the discrete solution to the biharmonic part of the discrete state and adjoint equations in (2.9a) and (2.9b).At each iteration of primal dual active set algorithm, the Newtons' method converges in ten iterations when the errors between final level and the penultimate level in Euclidean norm is less than 10 −9 .The primal dual active set algorithm terminates within four steps.
The numerical experiments are performed over the uniform and adaptive refinements.The uniform mesh refinement has been done by red-refinement criteria, where each triangle is subdivided into four sub-triangles by connecting the midpoints of the edges.The standard adaptive algorithm Solve-Estimate-Mark-Refinement [11,34] is used for the adaptive refinement, which is described in Section 6.Two examples are presented to illustrate the a priori and a posteriori reliability and efficiency estimates with = Ω so that C = I.The first example is considered over unit square domain where the solution of von Kármán equations is sufficiently smooth and the second example is over an L-shaped domain where the solution of von Kármán equations belongs to V ∩ 2+ (Ω) with ≈ 0.5445.

Uniform refinement
Example 6.1.(Convex Domain) Let the computational domain be Ω = (0, 1) 2 .The model problem is constructed in such a way that the exact solution is known.The data in the distributed optimal control problem are chosen as where Ψ = ( ¯ 1 , ¯ 2 ) and Θ = ( ¯ 1 , ¯ 2 ) denote the optimal state and adjoint variables.The source terms , and observation Ψ = ( ¯ ,1 , ¯ ,2 ) for Ψ are then computed using The relative errors and orders of convergence for the state, adjoint and control variables and the combined relative error and order of convergence are presented in Table 1.Since Ω is convex, Theorem 3.4 predicts linear order of convergence for the state and adjoint variables (resp.control variable) in the energy (resp. 2 ) norm.These theoretical rates of convergence are confirmed by the numerical outputs.
, ( ) where ≈ 0.5444837367 is a non-characteristic root of sin 2 ( ) = 2 sin 2 ( ), = 3  2 , and , ( )  Table 2 shows error estimates and the convergence rates of the state, adjoint and control variables.Since Ω is nonconvex, only suboptimal orders of convergence for the state and adjoint variables in the energy norm are obtained as predicted by Theorem 3.4.The numerical results show a better convergence rate for control which probably indicates that the numerical performance is carried out in the non-asymptotic region.

Adaptive mesh refinement
The standard adaptive algorithm: Solve-Estimate-Mark-Refine is used for the adaptive mesh-refinement.The total estimator 2 :=  1.This is a test case over the square domain with a smooth exact solution, performed to test the performance of the adaptive estimator for the uniform refinement.Table 3 depicts the convergence history of the estimators (defined in (4.2)) for the uniform refinements for the state, adjoint and control estimators.The combined error and estimator's convergence are also computed.It is observed that the individual errors and estimators as well as the combined error have linear order of convergence.Hence, the theoretical rates of convergence are confirmed by these numerical outputs.Non-convex Domain: This numerical experiment is performed over the non-convex domain (Example 6.2) with the exact solution has a singularity at the origin.The numerical experiment starts on the initial mesh with 24 triangles, and then adaptive refinements are carried out using Algorithm .Figure 1 shows that the significant adaptive refinement occurs near the control variable interface and the singularity point of the L-shaped domain.This is somewhat expected as the state and adjoint solutions have a singularity at the origin, and from Figure 2 it is observed that the control estimator dominates other estimators.This supports the efficiency of the adaptive estimator in the theoretical estimates obtained in the previous section.Figure 2 and Table 4 also indicate that the errors and estimators have optimal convergence in the adaptive refinement.Figure 3.( ) displays the convergence history of the total error and estimator; both achieve optimal convergence in adaptive refinement.Further, it can be observed that the adaptive refinements are doing better in terms of accuracy compared to the uniform refinements.Figure 3.( ) illustrates that reliability and efficiency constants are approaching a constant value with mesh refinement, which is numerical evidence for the efficiency and reliability of a posteriori estimator derived in the theory section.

Remark 4 . 1 .
( ) Note that the terms involoving P 0 in the reliability estimator of adjoint equations 2 , ΘM of (4.2d) are due to the combined effect of non-conformity of the method plus linear lower-order terms.( ) It is possible to avoid the terms involving P 0 in the reliability estimator2   , ΘM of (4.2d) which comes from 5 = NC ( ΨM , ( M − 1)Φ, ΘM ) in(4.

Proof of Theorem 5 . 1 .
Recall the definition of the complete estimator from (4.3).The summation over all the element and edges of the triangulation T , and the local efficiency results in Lemmas 5.2-5.

Figure 2 :
Figure 2: Convergence plot of the approximation errors and estimators with adaptive and uniform refinement for state, adjoint and control variables in Example 6.2 (State approximation (left), Adjoint approximation (Middle), Control approximation (right), Uni=Uniform refinement, Ada=Adaptive refinement).

Table 2 :
Errors and orders of convergence for state, adjoint and control variables in Example 6.2 Adaptive Mesh-refinement Algorithm Set the initial triangulation T 0 ; Set the maximum number of iteration Max ℓ ; while ℓ < Max ℓ do Solve: Compute the solution ( ΨM , ΘM , ¯ ℎ ) over the triangulation T ℓ using Newtons' method and primal dual active set strategy; Estimate: Compute the complete estimator 2 ℓ from (4.3); Mark: Mark a minimal subset M ℓ ⊂ T ℓ by Dörfler marking criteria;

Table 3 :
Estimator and order of convergence for state, adjoint and control variables in Example 6.1

Table 4 :
Errors and orders of convergence for state, adjoint and control variables with adaptive refinement in Example 6.2