We consider the following distributed optimal control problem governed by the diffusion equation with Neumann boundary condition:
Here, () is a bounded domain with boundary , is the outer unit normal to Ω and
denotes the average value of the function y over Ω (here and in the rest of the paper, is the Lebesgue measure of a set ). The cost functional, dependent on the state variable y and the control variable u, is given by
with . The desired state variable is chosen to satisfy . The source term also satisfies the zero average condition . The diffusion matrix is a measurable, bounded and uniformly elliptic matrix-valued function such that is symmetric for a.e. . Finally, the admissible set of controls is the non-empty convex set defined by
where a and b are constants in with (this condition is necessary to ensure that is not empty or reduced to ).
In this paper, we discuss the discretisation of control and state by the gradient discretisation method (GDM). The GDM is a generic framework for the convergence analysis of numerical methods (finite elements, mixed finite elements, finite volume, mimetic finite difference methods, etc.) for linear and non-linear elliptic and parabolic diffusion equations (including degenerate equations), the Navier–Stokes equations, variational inequalities, Darcy flows in fractured media, etc. See for example [16, 17, 18, 1, 22], and the monograph  for a complete presentation of the GDM. The contributions of this article are summarised now. Here, we
establish basic error estimates that provide convergence rate for all the three variables (control, state and adjoint) for low-order schemes under standard regularity assumptions for the pure Neumann problem, without reaction term. Given that the optimal control is approximated by piecewise constant functions, the convergence rates are optimal.
prove super-convergence result for post-processed optimal controls, state and adjoint variables.
establish a projection relation (see Lemma 3.10) between control and adjoint variables. This relation, which is non-standard since it has to account for the zero average constraints, is the key to prove the super-convergence result for all three variables.
design a modified active set strategy algorithm for GDM that is adapted to this non-standard projection relation.
discuss some numerical results that confirm the theoretical rates of convergence for conforming, non-conforming and mimetic finite difference methods.
The literature contains several contributions to numerical analysis for second-order distributed optimal control problems governed by diffusion equation with Dirichlet and Neumann boundary conditions (BCs) (we refer to [20, 30] and the references therein). For Dirichlet BCs, the super-convergence of post-processed controls for conforming finite element (FE) methods has been investigated in . Recently, this result was extended to the GDM in . Carrying out this analysis in the context of the GDM means that it readily applies to various schemes, including non-conforming finite elements and hybrid mimetic mixed schemes (HMM), which contains mixed-hybrid mimetic finite differences; and for these schemes, the analysis of  provides novel estimates. We refer to [8, 9, 10, 11, 12, 13] for an analysis of non-linear elliptic optimal control problems. Several works cover optimal control for second-order Neumann boundary value problems, albeit with an additional (linear or non-linear) reaction term which makes the state equation naturally well-posed, without zero average constraint, see [2, 3, 11, 26, 29]. To the best of our knowledge, the numerical analysis of pure Neumann control problems, without reaction term and thus with the integral constraint, is open even for finite element methods. Our results therefore seem to be new and, being established in the GDM framework, cover a range of numerical methods, including conforming Galerkin methods, non-conforming finite elements, and mimetic finite differences. Although done on the simple problem (1.1), the analysis uncovers some properties, such as the specific relation formula between the adjoint and control variables and a modified active set algorithm used to compute the solution of the numerical scheme. These have a wider application potential in optimisation problems involving an integral constraint. For example, in the model of [14, 31] describing the miscible displacement of one fluid by another in a porous medium, the pressure is subjected to an elliptic equation with homogeneous Neumann boundary conditions. In this equation, the source terms model the injection and production wells, and are typically the only quantities that engineers can adjust (to some extent). Hence, considering these source terms as controls of the pressure may lead to optimal control problems as in (1.1), with the exact same boundary conditions and integral constraints on the control terms.
The paper is organised as follows. Section 2 recalls the GDM for elliptic problems with Neumann BCs and the properties needed to prove its convergence. Some classical examples of GDM are presented in Section 2.2. Section 3 deals with the GDM for the optimal control problem (1.1). The basic error estimates and super-convergence results are presented in Sections 3.2 and 3.3. The first super-convergence result provides a nearly quadratic convergence rate for a post-processed control. Under an stability assumption of the GDM, which stems for most schemes from the quasi-uniformity of the mesh, the second super-convergence theorem establishes a full quadratic super-convergence rate. Discussions on post-processed controls and the projection relation between control and proper adjoint are presented in Section 3.4. The active set strategy is an algorithm to solve the non-linear Karush–Kuhn–Tucker (KKT) formulation of the optimal control problem . Section 4.1 presents a modification of this algorithm that accounts for the zero average constraint on the control. This modified active set algorithm also automatically selects the proper discrete adjoint whose projection provides the discrete control variable. In Section 4.2, we present the results of some numerical experiments. The paper ends with an appendix, Section A, where we prove the results stated in Sections 3.2 and 3.3.
Before concluding this introduction, we discuss the optimality conditions for (1.1). For a given , there exists a unique weak solution of (1.1b)–(1.1c). That is, for , there exists a unique such that for all ,
The term is the state associated with the control u.
In the following, the norm and scalar product in (or for vector-valued functions) are denoted by and . The convex control problem (1.1) has a unique solution and there exists a co-state such that the triplet satisfies the KKT optimality conditions [28, Chapter 2]:
Remark 1.1 (Choice of the Adjoint State).
Several co-states satisfy the optimality conditions (1.4), as is only determined up to an additive constant by (1.4). The same will be true for the discrete co-state, solution to a discrete version of these KKT equations. Establishing error estimates require the continuous and discrete co-states to have the same average. The usual choice is to fix this average as zero. However, for the control problem with pure Neumann conditions, this is not the best choice. Indeed, as seen in Lemma 3.10, establishing a proper relation between the control and co-state requires a certain zero average of a non-linear function of this co-state. A more efficient approach, that we will adopt, to fix the proper co-states is thus the following:
Design an algorithm (the modified active set algorithm of Section 4.1) that computes a discrete co-state with the proper condition, so that the discrete control can be easily obtained in terms of this discrete co-state.
Fix the average of the continuous co-state to be the same as the average of the discrete co-state obtained above.
As we will see, an algebraic relation between this and the continuous control can still be written, upon selecting a proper (but non-explicit) translation of .
Remark 1.2 (Zero Average Constraint on the Source Term and Desired State).
(i) If we consider (1.1) without the constraint on the source term, the set of admissible controls needs to be modified into
In this case, a simple transformation can bring us back to the case of a source term with zero average. Rewrite the state equation (1.1b) as with and . Then and , where
(ii) If the desired state is such that , then it is natural to select states y in (1.1) with the same average m (since the average of these states can be freely fixed, and the choice made in (1.1c) is arbitrary). This ensures the best possible approximation of the desired state . In that case, working with and instead of y and brings back to the original formulation (1.1) with a desired state having a zero average.
Remark 1.3 (Non-homogeneous BCs).
The study of the second-order distributed control problem (1.1) with non-homogeneous boundary conditions on (with ) follows in a similar way. In this case, the source terms and boundary condition are supposed to satisfy the compatibility condition
where is the trace operator and is the inner product in .
2 GDM for Elliptic PDE with Neumann BCs
The GDM consists in writing numerical schemes, called gradient schemes (GS), by replacing the continuous space and operators by discrete ones in the weak formulation of the problem [15, 16, 21]. These discrete space and operators are given by a gradient discretisation (GD).
2.1 Gradient Discretisation and Gradient Scheme
A notion of gradient discretisation for Neumann BCs is given in [15, Definition 3.1]. The following extends this definition by demanding the existence of the element and is always satisfied in practical applications. This existence ensures that the zero average condition can be put in the discretisation space or in the bilinear form as for the continuous formulation, see Remark 2.2.
Definition 2.1 (Gradient Discretisation for Neumann BCs).
A gradient discretisation (GD) for homogeneous Neumann BCs is given by such that:
is a finite-dimensional vector space on .
and are linear mappings.
is a norm on .
There exists such that on Ω and on Ω.
The flexibility of the GDM analysis framework comes from the wide possible range of choices for the gradient discretisation . Each of these choices correspond to a particular numerical scheme (see Section 2.2 for a few examples). The space represents the degrees of freedom (unknowns) of the method; a vector in gathers values for such unknowns. The operators and reconstruct, from a set of values of these unknowns, a scalar (respectively, vector) function on the entire set Ω. The scalar function is suppose to play the role of the solution/test functions itself in the weak formulation of the PDE; the vector-function, reconstructed “gradient”, is used in lieu of the gradients of these solution/test functions. Performing these substitutions in the weak formulation leads to a finite-dimensional system of equations (on the unknowns), which is dubbed the gradient scheme corresponding to the gradient discretisation .
If is such that , the weak formulation of the Neumann boundary value problem
is given as follows:
As explained above, a gradient scheme for (2.2) is then obtained from a GD by writing the weak formulation (2.3) with the continuous spaces, functions and gradients replaced with their discrete counterparts:
As for the continuous formulation (2.3), using the element actually enables us to consider in (2.4) test functions in , rather than just . The simplest technique to achieve this is to use a quadratic penalty method [24, Chapter 11]. For any , (2.4) can be shown equivalent to
2.2 Examples of Gradient Discretisations
We briefly present here a few examples based on known numerical methods. We refer to  for a detailed analysis of these methods, and more examples of gradient discretisations. As demonstrated by these examples, the GDM cover a wide range of different numerical methods. This means that the analysis carried out in the GDM framework for the control problem in (1.1) readily applies to all these methods. In particular, this makes the control problem accessible to numerical schemes not usually considered but relevant to diffusion models, such as schemes applicable on generic meshes (not just triangular/quadrangular meshes) as encountered for example in reservoir engineering applications.
Let us consider a mesh of Ω. A precise definition can be found for example in [15, Definition 7.2] but, for our purpose, the intuitive understanding of as a partition of Ω in polygonal/polyhedral sets is sufficient.
Conforming Finite Elements.
The simplest gradient discretisation is perhaps obtained by considering conforming finite elements. The mesh is made of triangles (in 2D) or tetrahedra (in 3D), with no hanging nodes. Each is a vector of values at the vertices of the mesh (the standard unknowns of conforming finite elements), is the continuous piecewise linear function on the mesh which takes these values at the vertices, and . Then (2.4) is the standard finite element scheme for (2.2).
Non-conforming Finite Elements.
As above, the mesh is made of conforming triangles or tetrahedra. Each is a vector of values at the centers of mass of the edges/faces, is the piecewise linear function on the mesh which takes these values at these centers of mass, and is the broken gradient of . In that case, (2.4) gives the non-conforming finite element approximation of (2.2).
Mass-Lumped Non-conforming Finite Elements.
Still considering a conforming triangular/tetrahedral mesh, the space and gradient reconstruction are identical to those of the non-conforming finite elements described above, but the function reconstruction is modified to be piecewise constant. For each edge/face of the mesh, we consider the diamond around constructed from the edge/face and the one or two cell centers on each side (see Figure 1). Then, for , the reconstructed function is the piecewise constant function on the diamonds, equal to on for all edge/face .
Hybrid Mixed Mimetic Method (HMM).
We consider a generic mesh (not necessarily triangular/tetrahedral) with one point chosen in each cell such that K is strictly star-shaped with respect to ; see Figure 2 for some notations. A vector is made of cell and face values, and the operator reconstructs a piecewise constant function from the cell values: for any cell K, on K. The gradient reconstruction is built in two pieces: a consistent gradient constant over the cell and stabilisation terms constant over the half-diamonds (and akin to the remainders of first-order Taylor expansions between the cell and face values). For any cell K and any face of K, we set
where is the orthogonal distance between and , is the center of mass of , is the outer normal to K on and, denoting by the set of faces of K,
Once used in the gradient scheme (2.4), this HMM gradient discretisation gives rise to a numerical method that can be applied on any mesh (including with hanging nodes, non-convex cells, etc.). This scheme can also be re-interpreted as a finite volume method [15, Section 13.3].
2.3 Error Estimates for the GDM for the Neumann Problem
The accuracy of a gradient scheme (2.4) is measured by three quantities. The first one is a discrete Poincaré constant , which ensures the coercivity of the method:
The second quantity is the interpolation error , which measures what is called, in the GDM framework, the GD-consistency of :
Finally, we measure the limit-conformity of a GD by defining
where with being the normal trace of on .
By using these quantities, an error estimate can be established for GS. We refer to  for a proof of the following theorem. Here and in the rest of the paper,
Theorem 2.3 (Error Estimate for the GDM).
Remark 2.4 (Rates of Convergence).
For all classical low-order methods based on meshes (such as conforming and non-conforming finite element methods, finite volume methods, etc.), if A is Lipschitz continuous and , then estimates can be obtained for and (see ). Theorem 2.3 then gives a linear rate of convergence for these methods.
Note that Theorem 2.3 also holds if we replace the zero average condition on ψ and with
In this case, estimate (2.10) can be obtained by considering the translation of and ψ. Set and , where
and 1 is the constant function. Using Definition 2.1, we find , and . This gives
Applying Theorem 2.3,
The following stability result, useful to our analysis, is straightforward.
Proposition 2.6 (Stability of the GDM).
Let be a coercivity constant of A. If is the solution to the gradient scheme (2.4), then
3 GDM for the Control Problem and Main Results
This section starts with a description of GDM for the optimal control problem and is followed by the basic error estimates and super-convergence results in Sections 3.2 and 3.3. The super-convergence results for a post-processed control are presented. A discussion on post-processed controls and the projection relation between control and proper adjoint are presented in Section 3.4.
3.1 GDM for the Optimal Control Problem
Let be a GD as in Definition 2.1. The space is defined as the space of piecewise constant functions on a mesh of Ω. The space is a finite-dimensional subset of . A gradient scheme for (1.4) consists in seeking such that
As in the continuous KKT conditions (1.4), these equations do not define uniquely. One possible constraint that fixes a unique is described in Lemma 3.10. This particular choice ensures a simple projection relation between and .
Two projection operators play a major role throughout the paper: the orthogonal projection on piecewise constant functions on , namely and the cut-off function . They are defined as
3.2 Basic Error Estimate for the GDM for the Control Problem
To state the error estimates, we define the projection error by
The proofs of the following basic error estimates are provided in Section A. They follow by adapting the corresponding proofs in  to account for the pure Neumann boundary conditions and integral constraints.
Theorem 3.1 (Control Estimate).
Proposition 3.2 (State and Adjoint Error Estimates).
3.3 Super-Convergence for Post-Processed Controls
We make here the following assumptions.
Assumption A1 (Interpolation Operator).
For each , there exists such that:
If solves , and is the solution to the corresponding GS with
For any , it holds
The estimate holds for any .
Assumption A3 (Discrete Sobolev Imbedding).
For all , it holds
where is a Sobolev exponent of 2, that is, if , and if .
and . That is, is the set of cells where crosses at least one constraint a or b. For , we let . The space is the usual broken Sobolev space, endowed with its broken norm. Our last assumption is:
We have and .
Possible choices of the mapping , depending on the considered gradient discretisation (that is, the considered numerical method), is discussed in Remark 3.4 below. Note that Assumptions A1–A4 are similar to that in  with substituted by , and an additional average condition in Assumption A1. We also refer to  for a detailed discussion on Assumptions A1–A4.
For a detailed discussion on the post-processed controls, we refer the reader to Section 3.4.
Remark 3.4 (Choice of in Assumption A1).
When is a piecewise linear reconstruction, the super-convergence result (3.8) usually holds with . This is for example well known for conforming and non-conforming FE method. When is piecewise constant on for all , the super-convergence (3.8) requires to project the exact solution on piecewise constant functions on the mesh. This is usually done by setting for all and all (or, equivalently at order , with the center of mass of K). With this choice, the super-convergence result is known, e.g., for mixed/hybrid and nodal mimetic finite difference schemes (see [4, 19]). As a consequence:
For FE methods, .
For mimetic finite difference methods, for all .
For and the center of mass of K, let be the maximal radius of balls centred at and included in K. Fixing such that
we use the following extension of the notation (2.9):
Theorem 3.5 (Super-Convergence for Post-Processed Controls I).
Let be a GD and let be a mesh. Assume that
and let , be the post-processed controls defined by (3.11), where and are chosen such that
Then there exists C depending only on α in (1.2) such that
Theorem 3.6 (Super-Convergence for Post-Processed Controls II).
Then there exists C depending only on α and δ such that
Corollary 3.8 (Super-Convergence for the State and Adjoint Variables).
Let the assumptions of Theorem 3.5 be fulfilled. Then the following error estimates hold, with C depending only on α:
where and are defined as in Assumption A1, and .
3.4 Discussion on Post-Processed Controls
In this subsection, we present a detailed analysis of post-processed controls given by (3.11). This analysis is performed under the assumptions of Section 3.3, and by also assuming that for all (see Remark 2.4). We begin by stating and proving two lemmas which discuss projection relations between control and adjoint variables for the pure Neumann problem, both at the continuous level and at the discrete level. We then show that the post-processed controls remain close to their corresponding original controls, see (3.20) and (3.24). Hence, the super-convergence result makes sense: since is piecewise constant, it is impossible to expect more than approximation on the controls; but by “moving” these controls by a specific , we obtain computable post-processed controls that enjoy an convergence result.
Let and . Define by
where is given by (3.3). Set and . Then we have the following:
Γ is Lipschitz continuous.
, , and there is such that .
If , then for any compact interval Q in , there exists such that if with , then
As a consequence, the real number in item 2 is unique.
Item 1 is obvious since is Lipschitz continuous.
Let us now analyse the limits in item 2. Let be a sequence in such that as . By the definition of M, this implies a.e. on Ω. Let be bounded below by R and note that . Moreover, for , implies so
By Fatou’s lemma,
which gives . Since (because ), we infer that
and thus that . In a similar way, we deduce that .
The existence of such that then follows from the Intermediate Value Theorem and
We now assume that and we turn to item 3. For a.e. , , where is the characteristic function of . Define for all . We claim that
Θ is lower semi-continuous,
for all .
To prove that Θ is lower semi-continuous, let as . Since is lower semi-continuous on , we have, for all ,
Applying Fatou’s lemma,
It follows that Θ is lower semi-continuous. We now show that on the interval . Let . Then is an interval of positive length, since and . The set has a non-zero measure because and Ω is connected. To see this, let be the endpoints of I and assume that takes some values less than α on a non-null set, some values greater than β on a non-null set, but that is a null set. Then exactly takes the values α and β (outside a set of zero measure). Hence and should be constant, since Ω is connected, which is a contradiction. Thus, has a non-zero measure. Since , this gives .
Coming back to item 3, let Q be a compact interval in . We know that on Q and Θ is lower semi-continuous. Hence Θ reaches its minimum on Q and for some . Since a.e., we have a.e. on Q and, Γ being Lipschitz and , we infer
which establishes (3.15). The uniqueness of such that follows from this inequality, which shows that Γ is strictly increasing on . ∎
Lemma 3.10 (Projection Formulas for the Controls).
If is a co-state and is such that
then the continuous optimal control in (1.4) can be expressed in terms of the projection formula
If is a GD and is chosen such that
then the discrete optimal control in (3.1) is given by
Set . Clearly, . From the optimality condition for the control problem (1.4c), we deduce that
since . Set , i.e.,
It is then straightforward to see that , i.e., and (by the choice of ). Then, using the definitions of , and , since on and on ,
Recall that the optimality condition is nothing but a characterisation of the orthogonal projection of on and, as such, defines a unique element of . We just proved that satisfies this optimality condition, which shows that it is equal to . The proof of (3.16) is complete.
The second relation follows in a similar way by noticing that, since controls are piecewise-constants on , (3.1c) is equivalent to for all . We also notice that, by the definition of and assumption (3.17), . ∎
Since the discrete co-state is a computable quantity, its average is easier to fix than the average of the non-computable . Hence, the projection relation (3.18) is the most natural choice to express the discrete control in terms of the discrete adjoint variable. This is the choice made in the modified active set strategy presented in Section 4.1. Once this choice is made, since must have the same average as for defined in (3.11) to satisfy super-convergence estimates, it is clear that will not have a zero average in general. Hence, if we want to express the continuous control in terms of , we need to offset this by the correct , as stated in the lemma.
Lemma 3.12 (Stability of the Discrete States).
With (2.8), this implies .
Therefore, for , recalling the definition (2.11) of , we have
The result for the adjoint variable can be derived similarly and hence (3.19) follows. ∎
In the rest of this section, we establish estimates between the controls , and their post-processed versions , . These estimates justify that the post-processed controls are indeed meaningful quantities.
Let us now turn to estimating . The co-state in (1.4) is still taken such that
By the choice of , . From Lemma 3.10, the choice of shows that
Let be the solution to (3.1b) with source term (that is, the solution to the GS for equation (1.4b) satisfied by ) such that . Using the Lipschitz continuity of , the Cauchy–Schwarz inequality, the triangle inequality, Remark 2.5, Proposition 2.6, Assumption A2, Theorem 2.3 and Lemma 3.12, we obtain
Let be as in Lemma 3.9 for . Relation (3.23) shows that if h is small enough; hence, in this case, . There is therefore a compact interval Q in depending only on such that 0 and belong to Q. Without loss of generality, we can assume that . A use of Lemma 3.9 leads to , where . This implies , using (3.23) and the fact that . Combining this with (3.22), we infer that
4 Numerical Experiments
In this section, we first present the modified active set strategy. This is followed by results of numerical experiments for conforming, non-conforming and mimetic finite difference methods.
4.1 A Modified Active Set Strategy
The interest of choosing an adjoint given by (3.17) is highlighted in Lemma 3.10: we have the projection relation (3.18) between the discrete control and adjoint. Such a relation is at the core of the (standard) active set algorithm . For a detailed analysis of this method, we refer the reader to [5, 6, 27]. Here, we propose a modified active set algorithm that enforces the proper zero average condition, and thus the proper relation between discrete adjoint and control.
where is constant.
Set . As the original active set strategy , the modified active set strategy is an iterative algorithm. As initial guesses, two arbitrary functions, are chosen. In the nth step of the algorithm, we define the set of active and inactive restrictions by
then we terminate the algorithm. In this case, we notice that the relative difference between and is less than for all examples in Section 4.2. Else we find and solution to the system
The above algorithm consists of non-linear equations. It can however be approximated by a linearized system in the following way, thus leading to our final modified active set algorithm. Instead of solving (4.2), we solve
Note that (4.3c) can be re-written in the following more commonly used form:
where and denote the characteristic functions of the sets and , respectively.
In this subsection, we illustrate examples for the numerical solution of (1.1). We use three specific schemes for the state and adjoint variables: conforming finite element (FE) method, non-conforming finite element (ncFE) method and hybrid mimetic mixed (HMM) method. All three are GDMs with gradient discretisations with bounds on , order h estimate on , and satisfying Assumptions A1–A4, and (3.12) on quasi-uniform meshes; see [15, 20] and Remarks 2.4 and 3.4.
The control variable is discretised using piecewise constant functions on the corresponding meshes. The discrete solution is computed using the modified active set algorithm mentioned in Section 4.1 with zero as an initial guess for both u and μ. Here, and denote the average values of the computed control and the reconstructed state solution respectively. Let ni denote the number of iterations required for the convergence of the modified active set algorithm, and denote the numerical average of the source term f calculated using the same quadrature rule as in the implementation of the schemes, i.e.,
where denotes the center of mass of the cell K. This numerical average enables us to evaluate the quality of the quadrature rule for each mesh; in particular, since f has a zero average, any quantity of the order of can be considered to be equal to zero, up to quadrature error. The relative errors are denoted by
The data in the optimal control problem (1.1) are chosen as follows:
where is chosen to ensure that . The matrix-valued function is given by unless otherwise specified. The source term f and the desired state are then computed using
4.2.1 Example 1: , , ,
We here consider the computational domain . We have and, since is odd, . Integrating this relation over Ω shows that has a zero average and thus, by Lemma 3.10, that . We thus see that .
The discrete solution is computed on a family of uniform grids with mesh sizes , . Due to the symmetry of the mesh and of the solution, approximate solutions are also symmetric and thus have zero average at an order compatible with the stopping criterion in the active set algorithm (the discrete solutions of (3.1) are only approximated by this algorithm), see Table 1. As also seen in this table, the number of iterations of the modified active set algorithm remains very small, and independent on the mesh size. The error estimates and the convergence rates of the control, the post-processed control, the state and the adjoint variables are presented in Table 2. The numerical results corroborate Theorem 3.1, Theorem 3.6 and Corollary 3.8.
For comparison, we compute the solutions of the nc finite element method on the same grids. As for conforming FE, the symmetry of the problem ensures that the approximation solutions have a zero average at an order dictated by the stopping criterion used in the active set algorithm. The results in Tables 3 and 4 are similar to those obtained with the conforming FE.
This scheme was tested on a series of regular triangular meshes from  where the points (see [17, Definition 2.21]) are located at the center of mass of the cells. These meshes are no longer symmetric and thus the symmetry of the approximate solution is lost. Zero averages are thus obtained up to quadrature error, see Table 5. It has been proved in [7, 19] that the state and adjoint equations enjoy a super-convergence property in norm for such a sequence of meshes; hence, as expected from Theorem 3.6, so does the scheme for the entire control problem after post-processing of the control. The errors in the energy norm and the norm, together with their orders of convergence, are presented in Table 6.
For all three methods (conforming FE, nc FE and HMM), the theoretical rates of convergence are confirmed by the numerical outputs. Without post-processing, an convergence rate is obtained on the controls, which validates Theorem 3.1. With post-processing of the controls, the order of convergence of Theorem 3.6 is recovered. We also notice that the super-convergence on the state and adjoint stated in Corollary 3.8 is confirmed, provided that the exact state and adjoint are properly projected (usage of the functions and in and ).
As seen in Table 5, the modified active set algorithm converges in very few iterations if . We however found that, if , the modified active set algorithm no longer converges. Further work will investigate in more depth the convergence analysis of the modified active set algorithm, to understand better its dependency with respect to ρ.
4.2.2 Example 2: , , , ,
In this subsection, we present some numerical results for the control problem defined on the unit square domain and . As explained in Example 1, and imply .
We provide in Table 7 the details of active set algorithm for the conforming finite element method. As expected, the symmetries of the problem provide approximate solutions with a nearly perfect average. For such grids, we obtain super-convergence result for the post-processed control. The errors between the true and computed solutions are computed for different mesh sizes and presented in Table 8. They still follow the expected theoretical rates, and the number of iterations of the active set algorithm remain small.
4.2.3 Example 3: , , ,
In this case, since is no longer odd, no longer has a zero average and, to compute , we need to find such that
This can be found by a bisection method, by computing the averages on a very thin mesh and bisecting until we find a proper . Using a mesh of size , we find .
The numerical results obtained using conforming finite element method are shown in Tables 13 and 14, respectively. Since there is a loss of symmetry, the approximate solutions have zero averages only up to quadrature error (compare and in Table 13). Here, we observed that the modified active set algorithm converges only when . When it does, though, the number of iterations remain very small. As in Examples 1 and 2, the theoretical rates of convergence are confirmed by these numerical outputs.
The proofs of error estimates for control, state and adjoint variables are obtained by modifying the proofs of the corresponding results in . For the sake of completeness and readability, we provide here detailed proofs, highlighting in chosen places where modifications are required due to the pure Neumann boundary conditions (which mostly amount to making sure that certain averages have been properly fixed).
Proof of Theorem 3.1.
Define the following auxiliary discrete problem: Seek
where the co-state is chosen such that
For Neumann boundary conditions, this particular choice is essential as it ensures that can be used as a test function in (3.1a) and (A.1a). Recalling that is the orthogonal projection on piecewise constant functions on , we obtain . Also, for and , and, using (1.3),
Adding these two inequalities yields
By the orthogonality property of , we have and . Therefore, the first term in the right-hand side of (A.2) can be re-cast as
By the Cauchy–Schwarz inequality, the first term on the right-hand side of (A.3) is estimated as
Equation (A.1b) shows that is the solution of the GS corresponding to the adjoint problem (1.4b), whose solution is . Therefore, using the fact that (note that the specific relation between the continuous and discrete co-states is essential here), by Theorem 2.3,
Hence, using the Cauchy–Schwarz inequality,
Using the definitions of , and the fact that , we find that
Choose in the above equality and use it in (A.7) to obtain
As a consequence,
Proof of Proposition 3.2.
An application of triangle inequality yields
Proof of Theorem 3.5.
Consider the auxiliary problem defined by: For , let solve
where is given by (A.1a) with replaced by g. We fix by imposing . This choice is dictated by the pure Neumann boundary condition and will be essential.
For , let be the centroid (centre of mass) of K. A standard approximation property (see e.g. [19, Lemma A.7] with ) yields
Define and a.e. on Ω by: For all and all , and . From (3.11) and the Lipschitz continuity of , we obtain
We now estimate the last term in this inequality. Use the definitions of , and the fact that
Plugged into (A.18), this estimate yields
Step 2: Estimate of . Subtract equations (A.15) satisfied by and to obtain, for all ,
Set , use the orthogonality of , the Cauchy–Schwarz inequality and Assumption A2 to infer
Consider now the last term on the right-hand side of (A.23). For any , we have on K, on K, or, by (3.16), on K. Hence, on K, or . This leads to on . Use (A.16), the definition of , the fact that and (A.24) to obtain
Since , and are continuous at the centroid , we can choose and on K). All the involved functions being constants over K, this gives
Integrate over K and sum over to deduce
Choose in the discrete optimality condition (3.1c) to establish
Add the above two inequalities to obtain
By the Cauchy–Schwarz inequality, the triangle inequality and the notations in (A.17), we obtain
Step 4: Conclusion. A use of (1.2) and the fact that is optimal leads to
where is the solution to the state equation (1.1b) with . Hence,
Note that . Therefore,
Proof of Theorem 3.6.
The proof of this theorem is identical to the proof of Theorem 3.5, except for the estimate of . This estimate is the only source of the power (instead of 2), and the only place where we used Assumption A3, here replaced by (3.12). Recall Assumption A4 and use (3.12) in (A.21) satisfied by to write
The rest of the proof follows from this estimate. ∎
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About the article
Published Online: 2017-12-05
Published in Print: 2018-10-01
Funding Source: Australian Research Council
Award identifier / Grant number: DP170100605
Award identifier / Grant number: SR/S4/MS/808/12
The first author acknowledges the funding support from the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (project number DP170100605). The second and third authors acknowledge the funding support from the DST project SR/S4/MS/808/12.