JOHANNES KEPLER UNIVERSITY LINZ. Multiharmonic Finite Element Analysis of a Time-Periodic Parabolic Optimal Control Problem

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1 JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics Multiharmonic Finite Element Analysis of a Time-Periodic Parabolic Optimal Control Problem Ulrich Langer Institute of Computational Mathematics, Johannes Kepler University Altenberger Str. 69, 44 Linz, Austria Monika Wolfmayr Institute of Computational Mathematics, Johannes Kepler University Altenberger Str. 69, 44 Linz, Austria NuMa-Report No January 213 A 44 LINZ, Altenbergerstraße 69, Austria

2 Technical Reports before 1998: Hedwig Brandstetter Was ist neu in Fortran 9? March G. Haase, B. Heise, M. Kuhn, U. Langer Adaptive Domain Decomposition Methods for Finite and Boundary Element August 1995 Equations Joachim Schöberl An Automatic Mesh Generator Using Geometric Rules for Two and Three Space August 1995 Dimensions Ferdinand Kickinger Automatic Mesh Generation for 3D Objects. February Mario Goppold, Gundolf Haase, Bodo Heise und Michael Kuhn Preprocessing in BE/FE Domain Decomposition Methods. February Bodo Heise A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element February 1996 Discretisation Bodo Heise und Michael Jung Robust Parallel Newton-Multilevel Methods. February Ferdinand Kickinger Algebraic Multigrid for Discrete Elliptic Second Order Problems. February Bodo Heise A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element Discretisation. May Michael Kuhn Benchmarking for Boundary Element Methods. June Bodo Heise, Michael Kuhn and Ulrich Langer A Mixed Variational Formulation for 3D Magnetostatics in the Space H(rot) February 1997 H(div) 97-2 Joachim Schöberl Robust Multigrid Preconditioning for Parameter Dependent Problems I: The June 1997 Stokes-type Case Ferdinand Kickinger, Sergei V. Nepomnyaschikh, Ralf Pfau, Joachim Schöberl Numerical Estimates of Inequalities in H 1 2. August Joachim Schöberl Programmbeschreibung NAOMI 2D und Algebraic Multigrid. September 1997 From 1998 to 28 technical reports were published by SFB13. Please see From 24 on reports were also published by RICAM. Please see For a complete list of NuMa reports see

3 MULTIHARMONIC FINITE ELEMENT ANALYSIS OF A TIME-PERIODIC PARABOLIC OPTIMAL CONTROL PROBLEM U. LANGER AND M. WOLFMAYR Abstract. This paper presents the multiharmonic analysis of a distributed parabolic optimal control problem in a time-periodic setting. We prove the existence and uniqueness of the solution of some weak space-time variational formulation for the parabolic time-periodic boundary value problem appearing in the constraints for the optimal control problem. Since the cost functional is quadratic, the optimal control problem is uniquely solvable as well. In order to solve the optimal control problem, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations which decouples into smaller systems. We construct preconditioners for these systems which yield robust convergence rates and optimal complexity for the preconditioned minimal residual method. All systems can be solved totally in parallel. Furthermore, we present a complete analysis for the error introduced by the multiharmonic finite element discretization as well as some numerical results confirming our theoretical findings. 1. Introduction This paper is devoted to distributed optimal control problems for parabolic timeperiodic boundary value problems in bounded two- or three-dimensional Lipschitz domains. Such kind of problems can be found not only in electromagnetics but also in other technical applications, see, e.g., [1, 37]. Let us mention that in two space dimensions eddy current problems arising frequently in electromagnetics turn to parabolic problems. Time-periodic parabolic problems and their numerical solution were considered in [17, 38, 39, 28]. The unique solvability of parabolic initialboundary value problems as well as of the parabolic problems in the time-periodic case is discussed in [42, 43]. In our work, we prove the unique solvability of a parabolic time-periodic boundary value problem in a special variational setting after introducing function spaces and formulating variational problems in the spirit of [27]. There are many papers on distributed optimal control problems for parabolic initial-boundary value problems, see, e.g., the recent paper [14] as well as the books [36, 9] and the references therein, but there are only a few papers on the time-periodic case, see [1, 19, 2] and [23, 22, 21]. The latter group of papers is devoted to the optimal control of time-periodic eddy current problems. In [1], the authors present a nested multigrid method for solving time-periodic parabolic optimal control problems. Our work considers the multiharmonic finite element method for solving these kind of problems, and we compare our numerical results to those obtained in [1]. The multiharmonic or the harmonic-balanced finite element method was successfully used for simulating electromagnetic devices which can be described by the eddy current approximation to Maxwell s equations, see [41, 31, 13, 16, 5, 6, 7, 12, 11]. Later the multiharmonic finite element method has been applied to time-periodic parabolic optimal control problems [19, 2, 26] and to time-periodic eddy current optimal control problems [23, 22, 21]. There are a couple of parameters involved in the time-periodic parabolic optimal control problem, e.g. the regularization or cost parameter as well as parameters which correspond to the conductivity and the reluctivity in practical applications in electromagnetics. Besides the discretization error analysis, the construction of fast solvers, which are 1

4 2 U. LANGER AND M. WOLFMAYR robust with respect to these bad parameters, are an important issue and a hot research topic during the last couple of years, see [18, 34, 44, 29, 32, 26]. In this paper, we provide a complete analysis of a time-periodic parabolic distributed optimal control problem and its multiharmonic finite element discretization. The discretization error analysis and the construction of robust solvers for the discrete optimality system are further contributions of this paper. More precisely, in Section 2, we state our time-periodic parabolic optimal control problem, investigate the unique solvability of the state equations in a special variational setting yielding the unique solvability of the optimal control problem. Finally, we derive the optimality system that we have to solve. Section 3 is devoted to the discretization of the optimality system by means of the multiharmonic finite element method. We expand the known desired state as well as the unknown state and co-state of the reduced optimality system into Fourier series and truncate them. The Fourier coefficients are then approximated by the finite element method. Due to the linearity and the orthogonality of the cosine and sine functions, the whole system of linear algebraic equations decouples into smaller systems depending only on the Fourier coefficients with respect to each single mode. Following [2], we construct preconditioners for our linear systems which yield robust convergence rates and optimal complexity for the preconditioned minimal residual (MINRES) method in Section 4. The preconditioners are practically implemented by a special algebraic multilevel iteration (AMLI) method that was proposed and analyzed in [24]. A complete analysis of the discretization error introduced by the multiharmonic finite element method is provided in Section 5. In Section 6, we present and discuss several numerical experiments to demonstrate the robustness and efficiency of the multiharmonic finite element method for solving time-periodic parabolic distributed optimal control problems. Finally, we draw some conclusions and give an outlook on some future work in Section A Time-Periodic Parabolic Optimal Control Problem Let us denote the state of our optimal control problem by y and the control by u. The spatial domain R d is given by a bounded Lipschitz domain with the boundary Γ :=, where d = 2, 3. We mention that the numerical results presented in Section 6 were obtained for the case d = 2. Moreover, the space-time cylinder is denoted by := (, T ) and its mantle boundary by Σ T := Γ (, T ). In this paper, we consider the following time-periodic parabolic distributed optimal control problem: (1) min J (y, u), y,u where the cost functional is given by the relation J (y, u) = 1 T (2) [y(x, t) y d (x, t)] 2 dx dt + λ 2 2 T [u(x, t)] 2 dx dt subject to the time-periodic parabolic PDE (Partial Differential Equation) constraints σ(x) t y(x, t) (ν(x) y(x, t)) = u(x, t), (x, t), (3) y(x, t) =, (x, t) Σ T, y(x, ) = y(x, T ), x, with strictly positive and uniformly bounded coefficients σ and ν, i.e. < σ σ(x) σ and < ν ν(x) ν, x.

5 PARABOLIC CONTROL PROBLEM 3 The desired state y d is the given target that we try to reach via a suitable control u. The positive regularization parameter λ provides a weighting of the cost of the control in the cost functional J (, ). In this section, we start with the existence and uniqueness proof for weak solutions of the PDE constraint described by (3), which guarantees the existence of a solution operator mapping the control space into the state space. This yields the existence of a unique solution of the optimal control problem (1)-(3). After that we formulate the optimality system of our problem The PDE constraint. First, we define proper function spaces which we will need for weak reformulations of our time-periodic problems, see, e.g., [27]. Definition 1. The Sobolev space H 1, ( ) is defined by H 1, ( ) = {y L 2 ( ) : y [L 2 ( )] d }, where is the weak spatial gradient, and equipped with the norm ( T ( y H 1, ( ) = y(x, t) 2 + ) 1/2 y(x, t) 2 dx dt). The Sobolev space H 1,1 ( ) is defined by H 1,1 ( ) = {y L 2 ( ) : y [L 2 ( )] d, t y L 2 ( )}, where t denotes the weak time derivative, and equipped with the norm ( T ( y H 1,1 ( ) = y(x, t) 2 + y(x, t) 2 + t y(x, t) ) 1/2 2 dx dt). For k =, 1, the Sobolev space H 1,k ( ) is defined by H 1,k ( ) = {y H 1,k ( ) : y = on Σ T }, whereas the Sobolev space H 1,1,per ( ) is given by H 1,1,per ( ) = {y H 1,1 ( ) : y(x, ) = y(x, T ) for almost all x }. Moreover, the Sobolev spaces H,1 ( ) and H,1 per( ) are analogously defined as follows and H,1 ( ) = {y L 2 ( ) : t y L 2 ( )} H,1 per( ) = {y H,1 ( ) : y(x, ) = y(x, T ) for almost all x }. For ease of notation, we will use the symbols (, ) L 2 () and L 2 () as well as the symbols (, ) H1 () and H1 () for indicating both the scalar and the vector-valued case. We denote the L 2 -inner product by n (v, w) L 2 () = (v i, w i ) L 2 (), i=1 where v = (v 1,..., v n ) T and w = (w 1,..., w n ) T are vectors. The associated norm is given by v 2 L 2 () = (v, v) L 2 (). In order to derive the space-time variational formulation, we multiply the parabolic PDE with a test function v H 1,1,per ( ), integrate over the space-time cylinder, and after integration by parts, we obtain the following space-time variational

6 4 U. LANGER AND M. WOLFMAYR formulation of the PDE constraint: Given the right hand side u L 2 ( ), find y H 1,1,per ( ) such that ( σ(x) y ) (4) v + ν(x) y v dx dt = u v dx dt v H 1,1,per t ( ). Since all functions are at least from the space L 2 ( ), we can expand them into Fourier series. The Fourier series expansion in time, e.g., for y, is given by (5) y(x, t) = y(x) c + (yk(x) c cos(kωt) + yk(x) s sin(kωt)) with the Fourier coefficients (6) y c k(x) = 2 T T y(x, t) cos(kωt) dt and y s k(x) = 2 T T y(x, t) sin(kωt) dt, where T and ω = 2π/T denote the periodicity and the frequency, respectively. In the following, we will use the notation v k = (v c k, v s k) T, v k = ( v s k, v c k) T and v k = (( v c k) T, ( v s k) T ) T. Inserting the Fourier series ansatz into the variational formulation (4), exploiting the orthogonality of the functions cos(kωt) and sin(kωt) with respect to the scalar product (, ) L 2 (,T ), we arrive at the following variational formulation corresponding to every single mode k N: Given u k (L 2 ()) 2, find y k V := V V = (H 1 ()) 2 such that ( ν(x) yk (x) v k (x) + kωσ(x)y k (x) v k (x) ) (7) dx = u k (x) v k (x) dx for all v k V. In the case k =, we obtain the following variational formulation: Given u c L 2 (), find y c V = H 1 () such that (8) ν(x) y(x) c v(x) c dx = u c (x)v(x) c dx. The space V is defined by V = H 1 () = {y L 2 () : y L 2 () and y = on Γ}. The space V = (H 1 ()) 2 for the Fourier coefficients is equipped with the norm y k 2 H 1 () = y k 2 L 2 () + y k 2 L 2 (). Theorem 1. The variational problems (7) and (8) have a unique solution. Proof. The proof follows from varifying the assumptions of the Lax-Milgram lemma and applying it. In order to prove existence and uniqueness of the space-time variational problem (4), we firstly prove the existence of a unique solution of a weaker variational formulation of our parabolic PDE (3). For that, we have to define additional special functions spaces. Definition 2. The function spaces H, 1 2 ( ) and H 1, 1 2 ( ) are defined by and H, 1 2 (QT ) = {y L 2 ( ) : 1/2 t y L 2 ( ) < } H 1, 1 2 (QT ) = {y H 1, ( ) : 1/2 t y L 2 ( ) < }, respectively, where 1/2 t y L2 is defined in the Fourier space by the relation ( ) 1/2 t y 2 := L 2 ( ) y 2 := T kω y H, 2 1 ( ) 2 k 2 L 2 ().

7 PARABOLIC CONTROL PROBLEM 5 Hence, the corresponding inner product is analogously given by ( 1/2 t y, 1/2 t v ) L 2 ( ) := T kω(y 2 k, v k ) L 2 (), and, in the same way, we define Moreover, we use the notation ( 1/2 t y, 1/2 t v ) L 2 ( ) := T 2 ( σ 1/2 t y, 1/2 t v ) L 2 ( ) := T 2 ( σ 1/2 t y, 1/2 t v ) L 2 ( ) := T 2 kω(y k, v k ) L 2 (). kω(σy k, v k ) L2 () and kω(σy k, v k ) L 2 () for the σ-weighted counterparts. The space H 1, 1 2 ( ) is given by H 1, 1 2 ( ) = {y H 1, 1 2 (QT ) : y = on Σ T }. The seminorm and the norm of the space H 1, 1 2 ( ) are defined in the Fourier space as follows: y 2 := T y c 2 H 1, 1 L 2 2 () + T [kω y 2 k 2 L 2 () + y k 2 L 2 () ] and y 2 H 1, 1 2 := T ( y c 2 L 2 () + yc 2 L 2 () ) + T 2 [(1 + kω) y k 2 L 2 () + y k 2 L 2 () ]. Furthermore, it is easy to see that the identities ( 1/2 t y, 1/2 t v ) = ( L 2 ( ) t y, v ) and ( 1/2 L 2 ( ) t y, 1/2 t v ) = ( L 2 ( ) t y, v ) L 2 ( ) are valid for all y Hper(Q,1 T ) and v H, 1 2 ( ). Indeed, inserting the Fourier expansions t y(x, t) := [kω yk(x) s cos(kωt) kω yk(x) c sin(kωt)] and v (x, t) := [yk(x) s cos(kωt) yk(x) c sin(kωt)], into the scalar products, we immediately get the identities given above. Furthermore, we obviously have v L 2 ( ) = v L 2 ( ), and the orthogonality relations ( t y, y ) L 2 ( ) = and (y, y) L 2 ( ) = for all y Hper(Q,1 T ), as well as (9) ( 1/2 t y, 1/2 t y ) L 2 ( ) = and ( y, y ) L 2 ( ) = for all y H 1, 1 2 ( ), which are also valid for their σ- and ν-weighted counterparts. Now, we are in the position to state a more general variational formulation of our space-time variational problem (4): Given u L 2 ( ), find y H 1, 1 2 ( ) such that ( σ(x) 1/2 y 1/2 v ) (1) + ν(x) y v dx dt = u v dx dt t 1/2 t 1/2

8 6 U. LANGER AND M. WOLFMAYR for all test functions v H 1, 1 2 ( ), where all functions are given in their Fourier series expansion in time. More precisely, everything has to be understood in the sense of Definition 2. The following lemma provides the existence of a unique solution of variational problem (1) and serves as vehicle for the existence and uniqueness proof of the space-time variational problem (4) and, later, for the construction of preconditioners and the discretization error analysis. Lemma 1. The space-time bilinear form ( a(y, v) = σ(x) 1/2 y 1/2 v ) + ν(x) y v dx dt t 1/2 t 1/2 fulfills the following inf-sup and sup-sup conditions: (11) µ 1 y H 1, 1 2 sup v H 1, 1 2 a(y, v) v H 1, 1 2 for all y H 1, 1 2 with positive constants µ 1 and µ 2. µ 2 y H 1, 1 2 Proof. We start with the proof of the sup-sup condition. Using triangle and Cauchy- Schwarz inequality, we obtain the estimate ( a(y, v) = σ(x) 1/2 y 1/2 v ) + ν(x) y v dx dt t 1/2 t 1/2 1/2 y 1/2 v σ t 1/2 t dx dt + ν y v dx dt 1/2 σ 1/2 t y L2 1/2 ( ) t v + ν y L2 ( ) L 2 ( ) v L2 ( ) max{σ, ν} y H 1, 1 2 v H 1, 1 2 µ 2 y H 1, 1 2 v H 1, 1 2 with the constant µ 2 = max{σ, ν}. Next, we prove the inf-sup condition by choosing the test function v = y + y. Using the σ- and ν-weighted orthogonality relations according to (9) and Friedrichs inequality y L2 () c F y L2 (), we get ( a(y, y) = σ(x) 1/2 y 1/2 y ) + ν(x) y y dx dt = ν(x) y y dx dt t 1/2 t 1/2 Q T ν y 2 1 dx dt ν c 2 F + 1 y 2 H 1, ( ) and a(y, y ) = = ( σ(x) 1/2 y t 1/2 σ(x) 1/2 y t 1/2 1/2 ) y + ν(x) y y dx dt t1/2 1/2 y t 1/2 dx dt σ 1/2 t y 2 L 2 ( ). Altogether, we have a(y, y + y ) ν c 2 F + 1 y 2 H 1, ( ) + σ 1/2 t y 2 L 2 ( ) ν ( min{ c 2 F + 1, σ} y 2 H 1, ( ) + 1/2 t y ) 2 = µ L 2 ( ) 1 y 2 H 1, 1 2 ( ) with the constant µ 1 = min{ c 2 F ν +1, σ}. Theorem 2. The space-time variational problem (1) has a unique solution. Proof. The proof follows immediately from Lemma 1 and from applying Babuška- Aziz theorem which was established in [3] and [4].

9 PARABOLIC CONTROL PROBLEM 7 Equipped with Lemma 1, we can prove existence and uniqueness of the following space-time variational problem which is weaker than the space-time variational problem (4) and also weaker than (1): Given u L 2 ( ), find y H 1, ( ) such that (12) ( σ(x)y v ) + ν(x) y v dx dt = u v dx dt v H 1,1,per t ( ). In order to prove existence and uniqueness of (12), we reuse some ideas known from the analysis of parabolic initial boundary value problems in [27], and adapt them to the time-periodic case. Theorem 3. The space-time variational problem (12) has a unique solution. Proof. We start with the uniqueness proof. Let us assume that there are two different solutions y 1 and y 2 H 1, ( ), y 1 y 2, of the problem (12). We expand these two solutions into Fourier series in time, i.e. y 1 (x, t) = [y1k c (x) cos(kωt) + ys 1k (x) sin(kωt)], y 2 (x, t) = k= [y2k c (x) cos(kωt) + ys 2k (x) sin(kωt)], k= whose unique Fourier coefficients are given by and y c 1k (x) = 2 T y c 2k (x) = 2 T T T y 1 (x, t) cos(kωt) dt and y s 1k (x) = 2 T y 2 (x, t) cos(kωt) dt and y s 2k (x) = 2 T T T y 1 (x, t) sin(kωt) dt, y 2 (x, t) sin(kωt) dt. Since y 1 y 2, we have y1k l (x) yl 2k (x) for at least one k N and l {c, s}, and therefore y 1k y 2k. Let us fix one arbitrary k for which y1k l yl 2k. For this k, we define w l k := y l 2k yl 1k and w k = (w c k, w s k) T V = V V = (H 1 ()) 2. After inserting the Fourier series ansatz for y 1 and y 2 into the variational problem (12), the whole system decouples and we arrive at variational problems for the Fourier coefficients with respect to every single mode k, analogously as in (7) and (8). For the arbitrary but fixed k, we obtain the following variational problem: Find w k V such that σ(x)kωw k v k dx + ν(x) w k v k dx = } {{ } :=a k (w k,v k ) for all v k V. We choose the test function v k = w k + w k. Hence, we obtain σ(x)kω(w k w k + w k (w k ) ) dx + ν(x)( w k w k + w k w k ) dx =. From (w k ) = w k and w k w k = it follows that (σ(x)kωw k w k + ν(x) w k w k ) dx =.

10 8 U. LANGER AND M. WOLFMAYR The bilinear form a k (, ) is elliptic and bounded in the space V = V V = (H 1 ()) 2. Hence, from = a k (w k, w k + w k ) σkω w k w k dx + ν w k w k dx ( ) min{σkω, ν} w k w k + w k w k dx, } {{ } = w k 2 H 1 () it follows that w k = in V. Since k was arbitrary, all Fourier coefficients of y 1 and y 2 have to be the same, hence y 1 = y 2, which is a contradiction to our assumption at the beginning. Thus, the uniqueness of a solution y H 1, ( ) of the variational problem (12) is proven. Now we come to the existence proof. First, we want to prove that if v H 1,1,per ( ) then the truncated Fourier series v N converges weakly in H 1,1,per ( ) to v. For that we expand v into a Fourier series in time, i.e. v(x, t) = [vk(x) c cos(kωt) + vk(x) s sin(kωt)], k= where the Fourier coefficients vk l (x) with l {c, s} are from H1 (). Hence, the truncated Fourier series N v N (x, t) = [vk(x) c cos(kωt) + vk(x) s sin(kωt)] k= converges strongly in L 2 ( ) to v, i.e. It remains to show that and v N v in L 2 ( ) for N. v N v in L 2 ( ) for N v N t v t in L 2 ( ) for N. Due to v H 1,1,per ( ), the weak derivatives of v exist. Integration by parts and the periodicity of the test function ϕ Cper(, T ) yield v N (x, t) ϕ(t) t dt dx = v N (x, t) t ϕ(t) dt dx for all ϕ Cper(, T ). We pass N to the limit and obtain v N (x, t) ϕ(t) dt dx v(x, t) ϕ(t) dt dx. t t So, there exists a w L 2 ( ) such that v N (x, t) t w in L 2 ( ) for N. We define v t := w L2 ( ). Moreover, due to integration by parts it follows that v N ϕ(x) dx dt = v N ϕ(x) dx dt for all ϕ [C ()] d, and v N v in L 2 ( ) yields v N ϕ(x) dx dt v ϕ(x) dx dt.

11 PARABOLIC CONTROL PROBLEM 9 Hence, there exists a w [L 2 ( )] d such that v N w in [L 2 ( )] d for N and we define v := w [L 2 ( )] d. Altogether, it follows that v N v in H 1,1,per ( ). From Theorem 1 (especially from the orthogonalities of the cosine and sine functions) it follows that y N solves the variational problem T ( ) v T N σ(x)y N + ν(x) y N v N dx dt = u v N dx dt, t where u is expanded into a Fourier series in time and y N and v N are truncated Fourier series in time. Moreover, y N solves the variational problems (4) and (1) for all v N, and y N, v N H 1, 1 2 ( ). Choosing the test function v N = y N + yn and using Lemma 1, we get the estimates y N 2 1 a(y H 1, 2 1 N, y N + y ( ) µ N) = 1 T u (y N + yn ) dx dt 1 µ 1 1 µ 1 u L2 ( ) y N + y N L2 ( ) Hence, y N is bounded, i.e. 1 µ 1 u L 2 ( )2 y N L 2 ( ) 2 µ 1 u L2 ( ) y N H 1, 1 2 ( ). (13) Moreover, (14) y N H 1, 1 2 ( ) 2 µ 1 u L 2 ( ) <. y N L 2 ( ) y N H 1, ( ) y N H 1, 1 2 ( ). From estimate (13) it follows that hence y N y in H 1, 1 2 ( ) for N, y N y in H 1, ( ) for N. It remains to show that y solves our variational problem (12). Let us choose M N arbitrarily and N N with N M, where v M v in H 1,1,per ( ) and u N u in L 2 ( ). Inserting y N, u N and the arbitrary test function v M into the variational problem (12) yields T ( ) v T M σ(x)y N + ν(x) y N v M dx dt = u N v M dx dt. t Since the test function v M v in H 1,1,per ( ) for M, T ( ) v σ(x)y N t + ν(x) y N v dx dt = T u N v dx dt. Now we pass N to the limit. Since u N u in L 2 ( ) and y N y in H 1, ( ), we finally get T ( σ(x)y v ) T + ν(x) y v dx dt = u v dx dt, t which means that y H 1, ( ) solves the variational problem (12).

12 1 U. LANGER AND M. WOLFMAYR Remark 1. Together with (13) it follows the existence and uniqueness of a solution of the variational problem (12) in the space H 1, 1 2 ( ) as well. The variational problems (1) and (12) are equivalent. Remark 2. If we assume that (ν y) L 2 ( ), then y H 1,1,per ( ). Hence, we obtain immediately existence and uniqueness of the variational problem (4) and we derive naturally the periodicity condition for y in a weak sense. More precisely, from (ν y) L 2 ( ) and integration by parts of (12) it follows that σ(x)y v t dx dt = (ν(x) y) v dx dt + u v dx dt for all test functions v H 1,1,per ( ). Hence, the weak time derivative of y is defined, i.e. σ(x)y v dx dt = ( (ν(x) y) + u) v dx dt t }{{} =:σ(x) y t with σ y t L 2 ( ). So the problem (12) together with (ν y) L 2 ( ) is equivalent to σ(x) y v dx dt σ(x) (y(x, T )v(x, T ) y(x, )v(x, )) dx t + (ν(x) y) v dx dt = u v dx dt for all v H 1,1,per ( ). Since v is periodic, σ(x) y v dx dt σ(x)(y(x, T ) y(x, ))v(x, ) dx t + (ν(x) y) v dx dt = u v dx dt and ( σ(x) y ) t (ν(x) y) u v dx dt = σ(x)(y(x, T ) y(x, ))v(x, ) dx for all v H 1,1,per ( ). Choose v H 1,1,per ( ) such that v(x, ) = for almost all x. Hence, ( σ(x) y ) t (ν(x) y) u v dx dt = for all v H 1,1,per ( ) and so σ(x) y t (ν(x) y) u = in L 2 ( ). It follows that σ(x)(y(x, T ) y(x, ))v(x, ) dx = for all v H 1,1,per ( ) and this yields for almost all x. y(x, T ) y(x, ) = Remark 3. Under classical regularity assumptions imposed on y, e.g., y C 2 ( ), and on the data u, σ, ν, e.g., u C( ), σ C(), ν C 1 (), it follows that y is the unique solution of the state equation (3).

13 PARABOLIC CONTROL PROBLEM 11 Theorem 2 implies the existence of a linear and continuous solution operator which uniquely assigns a state y H 1, 1 2 ( ) to every control u L 2 ( ), where the space H 1, 1 2 ( ) is compactly embedded in L 2 ( ). With the solution operator, the optimal control problem (1)-(3) can be rewritten as a reduced minimization problem. The existence of its unique solution can be analogously proven as, e.g., in [36] under the assumptions that y d L 2 ( ) and λ >. Note that here (3) has to be understood in a weak sense, more precisely, in the sense of (1) The optimality system. We want to formulate now the optimality system. Its solution is equivalent to the solution of the original optimal control problem (1)- (3). Let us denote the Lagrange multiplier by p. We choose the following Lagrange functional for our minimization problem: T ( L(y, u, p) := J (y, u) σ y t (ν y ) ) (15) u p dx dt. The three conditions (16) y L(y, u, p) =, u L(y, u, p) =, p L(y, u, p) =, are called optimality system and characterize a stationary point (y, u, p) of the Lagrange functional (15). Using the second condition, we eliminate the control u from the optimality system (16), i.e. (17) u = λ 1 p in. Note that the Lagrange multiplier p is also refered as the co-state. From (17) it appears very natural to choose y, p and also u all from the same space, see [2]. Moreover, we arrive at a reduced optimality system, written in its classical formulation as σ(x) t y(x, t) (ν(x) y(x, t)) = λ 1 p(x, t), (x, t), (18) y(x, t) =, (x, t) Σ T, y(x, ) = y(x, T ), x, σ(x) t p(x, t) (ν(x) p(x, t)) = y(x, t) y d(x, t), (x, t), p(x, t) =, (x, t) Σ T, p(x, T ) = p(x, ), x. The space-time variational formulation of (18) is obtained in the same way as for the PDE constraint (1) in Subsection 2.1 and is stated as the following: Given the desired state y d L 2 ( ), find y and p from H 1, 1 2 ( ) such that ( y v ν(x) p v + σ(x) 1/2 p 1/2 v ) dx dt = y t 1/2 t 1/2 d v dx dt, Q (19) T ( ν(x) y q + σ(x) 1/2 y 1/2 q ) t 1/2 t + 1/2 λ 1 p q dx dt = for all test functions v, q H 1, 1 2 ( ), where all functions are given in their Fourier series expansion in time according to Definition 2.

14 12 U. LANGER AND M. WOLFMAYR 3. Multiharmonic Finite Element Discretization In order to solve our optimal control problem (1)-(3), we solve the optimality system (19) by a multiharmonic finite element discretization. We approximate the desired state y d by truncating its Fourier series expansion and arrive at (2) y d (x, t) N [ydk c (x) cos(kωt) + ys dk (x) sin(kωt)] = y dn(x, t), k= where its Fourier coefficients are given by (21) y c dk (x) = 2 T T y d (x, t) cos(kωt) dt and y s dk (x) = 2 T T y d (x, t) sin(kωt) dt. We mention here that in general we have to compute the Fourier coefficients numerically, but for the moment we consider only the case where we can compute the Fourier coefficients exactly. In the case that y d has really a multiharmonic representation, we refer to [2]. We insert the truncated desired state and the Fourier series ansatz of the state y and the co-state p into the space-time variational formulation (19). From the orthogonality of the functions cos(kωt) and sin(kωt) it follows that it is sufficient to consider only the truncated Fourier series of y and p, i.e. N y(x, t) [y k(x) c cos(kωt) + yk(x) s sin(kωt)] = y N (x, t), k= (22) N p(x, t) [p c k(x) cos(kωt) + p s k(x) sin(kωt)] = p N (x, t). k= We arrive at the following system which has to be solved for every mode k = 1, 2,..., N: Find y k, p k V = V V = (H 1 ()) 2 such that (23) ( yk v k ν(x) p k v k + kωσ(x)p k v ) k dx = y dk v k dx, ( ν(x) yk q k + kωσ(x)y k q k + λ 1 ) p k q k dx = for all test functions v k, q k V = V V. In the case of k =, we obtain the following optimality system: Find y c, p c V = H 1 () such that (24) ( y c v c ν(x) p c v c ) dx = yd c vc dx, v c V = H 1 () ( ν(x) y c q c + λ 1 p c q c ) dx =, q c V = H 1 (). We want to approximate the unknown Fourier coefficients y k = (y c k, ys k )T, p k = (p c k, ps k )T V by finite element functions y kh = (y c kh, ys kh )T, p kh = (p c kh, ps kh )T V h = V h V h V. The space V h = V h V h is a finite element space, where V h = span{ϕ 1,..., ϕ n } with the standard nodal basis {ϕ i (x) = ϕ ih (x) : i = 1, 2,..., n h } and h denotes the usual discretization parameter such that n = n h = dimv h = O(h d ). We will use continuous, piecewise linear finite elements on triangles (2d) and tetrahedrons (3d) on a regular triangulation to construct the finite element subspace V h and its basis in our numerical experiments, see [1]. This yields the

15 PARABOLIC CONTROL PROBLEM 13 following linear system arising from the variational formulation (23): M h K h kωm h,σ y c k M h kωm h,σ K h y s (25) K h kωm h,σ λ 1 k M h p c = kωm h,σ K h λ 1 k M h p s k y c dk y s dk which has to be solved with respect to the nodal parameter vectors y j k = yj kh = (y j k,i ) i=1,...,n R n and p j = k pj = kh (pj k,i ) i=1,...,n R n of the finite element approximations n (26) y j kh (x) = n y j k,i ϕ i(x) and p j kh (x) = p j k,i ϕ i(x) i=1 to the unknown Fourier coefficients y j k (x) and pj k (x) with j {c, s}. The matrix M h, M h,σ and K h correspond to the mass matrix, the weighted mass matrix and the weighted stiffness matrix, respectively. Their entries are computed by the following formulas: M ij h = ϕ i ϕ j dx, with i, j = 1,..., n, whereas [ ] y c = y c dk dk ϕ j dx M ij h,σ = j=1,...,n σϕ i ϕ j dx, i=1, K ij h = ν ϕ i ϕ j dx, [ ] and y s = y s dk dk ϕ j dx. j=1,...,n In the case k =, we obtain the following linear system arising from the variational problem (24): ( ) ( ) ( ) Mh K h y c K h λ 1 y c (27) M h p c = d. From the solutions of the linear systems (25) and (27) we can easily reconstruct the multiharmonic finite element approximations of the state y(x, t) and the co-state p(x, t), i.e. N y Nh (x, t) = [ykh(x) c cos(kωt) + ykh(x) s sin(kωt)], k= (28) N p Nh (x, t) = [p c kh(x) cos(kωt) + p s kh(x) sin(kωt)]. k= We will present an error analyis for the complete discretization error between the unknown solution (y, p) and its multiharmonic finite element approximation (y Nh, p Nh ) in Section The inf-sup and sup-sup Condition and Robust Preconditioning The linear system (25) as well as the system (27) are saddle point problems of the form ( ) ( ) ( ) A B T y f (29) =. B C p }{{}}{{} =:A =:f In the case k = 1,..., N, we have that ( ) ( ) Mh Kh kωm A :=, B := h,σ, C := λ 1 A M h kωm h,σ K h

16 14 U. LANGER AND M. WOLFMAYR and f := ( y c dk y s dk ), y := ( y c k y s k ) ( p c, p := k p s k ). Saddle point problems of the form (29) can be solved by a preconditioned MINRES method, see [3]. Hence, it is crucial to construct preconditioners, which yield robust and fast convergence for the preconditioned MINRES method. In Subsection 4.1 and 4.2, we follow [2] by constructing such preconditioners Robust Preconditioning. In [2], the authors constructed preconditioners for the case that the parameter σ is constant, hence M h,σ = σm h, by following the strategy presented in [44]. The idea is to construct two preconditioners which yield robust convergence rates for the preconditioned MINRES method and apply the operator interpolation theorem, which is based on the construction of intermediate spaces via the so called real interpolation method. The ideas of the real method (Jand the K-method) are due to Lions and Peetre and the theory of the real method is developed e.g. in [8], see also [2]. Finally, in [2], the authors arrive at the following preconditioner for the system (29) in the case where σ is constant: (3) P = D D λ 1 D λ 1 D with the block diagonal matrices D := λk h + (kωσ λ + 1)M h. This yields the following condition number estimate: κ P (P 1 A) c/c with constants c, c independent of all involved parameters. In the case that the parameter σ is only piecewise constant, hence M h,σ σm h and (31) A := M h K h kωm h,σ M h kωm h,σ K h K h kωm h,σ λ 1 M h kωm h,σ K h λ 1 M h we cannot apply the operator interpolation theory anymore. According to [2], we take the block diagonal preconditioner P and replace σm h by M h,σ in (3). Hence, we arrive at the new preconditioner P but with D := λk h + kω λm h,σ + M h. The goal is to obtain robust norm estimates for the preconditioned system matrix P 1 A. In [2], the authors were able to verify the assumptions of Babuška-Aziz theorem, which are equivalent to the norm estimates for the precondioned system matrix. In the following, we will summarize their results. We define the following bilinear form arising from the optimality system (23): B((y k, p k ), (v k, q k )) := y k v k ν p k v k + kωσp k v k dx (32) + ν y k q k + kωσy k q k + λ 1 p k q k dx. Hence, the variational problem (23) now reads as follows: Find (y k, p k ) V 2 = V V = (H 1 ()) 4 such that (33) B((y k, p k ), (v k, q k )) = y dk v k dx for all test functions (v k, q k ) V 2.,

17 PARABOLIC CONTROL PROBLEM 15 We define the following inner products and associated norms inspired by the preconditioner P, which we need in order to prove the assumptions of Babuška- Aziz theorem. We first define an inner product in V = (H 1 ()) 2 by (34) (y k, v k ) V = λ (ν y k, v k ) L 2 () + kω λ (σy k, v k ) L 2 () + (y k, v k ) L 2 () with the associated norm (35) y k 2 V = λ (ν y k, y k ) L2 () + kω λ (σy k, y k ) L2 () + y k 2 L 2 (), which differs from the standard H 1 -norms. Next, we define an inner product in V 2 = (H 1 ()) 4 by (36) with the associated norm (37) ((y k, p k ), (v k, q k )) P = (y k, v k ) V + λ 1 (p k, q k ) V (y k, p k ) 2 P = y k 2 V + λ 1 p k 2 V. The following theorem verifies the assumptions of Babuška-Aziz theorem, which are equivalent to the norm estimates for our preconditioned system P 1 A. Moreover, it follows the existence and uniqueness of the solution of the variational problem (33) due to the theorem of Babuška-Aziz. Theorem 4. The following inequalities (38) c (y k, p k ) P B((y sup k, p k ), (v k, q k )) c (y (v k,q k ) V (v 2 k, q k ) k, p k ) P P hold for all (y k, p k ) V 2 with constants c = 1/ 3 and c = 1. Proof. See [2]. In the discrete case, we can repeat the proof of Theorem 4 step-by-step replacing V 2 by V 2 h, and we finally arrive at the same inequalities with the same constants. In matrix-vector notation, we have proven the inequalities (39) c x P sup z R 4n (Ax, z) z P c x P x R 4n implying the condition number estimate (4) κ P (P 1 A) := P 1 A P A 1 P P c/c = 3, which yields the following robust convergence rate of the preconditioned MINRES method: q = κ P(P 1 A) κ P (P 1 A) , see [2] and the references therein, e.g., [15]. The preconditioner for the discretized system (27) in the case of k = is analogously determined (cf. [2]). It follows the preconditioner ( ) D (41) P = λ 1 D with the diagonal block matrix D := M h + λk h. following condition number estimate: (42) κ P (P 1 A) 2, In this case, we obtain the which provides a robust convergence rate of the preconditioned MINRES method with 2 1 q , 2 + 1

18 16 U. LANGER AND M. WOLFMAYR see [2]. In a nutshell, we have designed preconditioners for the linear systems (25) and (27) corresponding to the modes 1 k N and k =, respectively, providing robust convergence rates for solving the preconditioned system by the preconditioned MINRES method Practical Implementation. In practical applications, for a large number of degrees of freedom, it is not very efficient to invert the preconditioners P in (3) of the discretized problem (25) for the case < k N and P in (41) of the problem (27) for the case k = exactly. Hence, it is important to replace the diagonal blocks D = λk h + kω λm h,σ + M h of the preconditioner P and the diagonal blocks D = M h + λk h of the preconditioner P by diagonal blocks D, which are spectrally equivalent to D, robust, symmetric positive definite and more cost efficient. Such practical preconditioners P can be obtained by various methods like DD (domain decomposition), AMG (algebraic multigrid) or AMLI (algebraic multilevel iteration) methods, see, e.g., [35, 25, 4, 33]. Hence, we get a new practical preconditioner, e.g., for the case < k N, i.e. (43) P = D D λ 1 D λ 1 D with c D D D cd D leading to the following robust condition number estimate: (44) κ P( P 1 A) κ P (P 1 A) (c D /c D ), where κ P (P 1 A) can be estimated by (4) for < k N and by (42) for k =. In Section 6, we will present some numerical results using an AMLI preconditioner proposed in [24]. 5. Discretization Error Analysis For the complete error analysis, we have to define norms in certain function spaces inspired by the P-norm and similar to the definitions (34)-(37). Definition 3. We define the following function spaces: where V := H 1, 1 2 ( ), V 1 := (H, 1 2 ) 1 ( ) H,1 per( ), (H, 1 2 ) 1 ( ) := {v H, 1 2 (QT ) : v H, 1 2 (QT ), v = on Σ T } with corresponding norms y 2 V = y 2 L 2 ( ) + λ (ν y, y) L 2 ( ) + λ ( σ 1/2 t y, 1/2 t y ) L 2 ( ), y 2 V 1 = y 2 L 2 ( ) + 1/2 t y 2 L 2 () + λ (ν y, y) L 2 ( ) + λ ( ν 1/2 t y, 1/2 t y ) L 2 ( ) + λ ( σ t y, t y ) L 2 ( ),

19 PARABOLIC CONTROL PROBLEM 17 all defined in the Fourier space according to Definition 2, i.e. y 2 V = T ( y c 2 L 2 () + λ(ν y, c y) c L 2 ()) + T [ y 2 k 2 L 2 () + λ(ν y k, y k ) L2 () + λkω(σy k, y k ) L2 ()], y 2 V 1 = T ( y c 2 L 2 () + λ(ν y, c y) c L2 ()) + T [(1 + kω) y 2 k 2 L 2 () + λ(1 + kω)(ν y k, y k ) L 2 () + λ(kω) 2 (σy k, y k ) L 2 ()]. Moreover, we define the following vector-valued norms as well: (y, p) 2 P = y 2 V + λ 1 p 2 V (y, p) 2 P 1 = y 2 V 1 + λ 1 p 2 V 1. Remark 4. Note that the V-norm defined in (35) as well as the P-norm defined in (37) correspond to the V -norm and the P -norm, respectively, for a single mode k, i.e. y 2 V = T y c 2 V + T 2 y k 2 V (y, p) 2 P = T (y c, p c ) 2 P + T 2 (y k, p k ) 2 P. The complete discretization error between the exact solution of the variational problem (19) and its multiharmonic finite element approximation is given by (45) (y, p) (y Nh, p Nh ) P, where the exact solution is expanded into Fourier series, i.e. y(x, t) = y c (x) + p(x, t) = p c (x) + [yk(x) c cos(kωt) + yk(x) s sin(kωt)], [p c k(x) cos(kωt) + p s k(x) sin(kωt)], and their multiharmonic finite element approximations are given by y Nh (x, t) = y c h(x) + p Nh (x, t) = p c h(x) + N [ykh(x) c cos(kωt) + ykh(x) s sin(kωt)], N [p c kh(x) cos(kωt) + p s kh(x) sin(kωt)]. Using triangle inequality, we can split the discretization error (45) into two parts, i.e. (y, p) (y Nh, p Nh ) P (y, p) (y N, p N ) P }{{} + (y N, p N ) (y Nh, p Nh ) P. }{{} discretization error in N discretization error in h 5.1. Discretization error in N. The following theorem provides an estimate for the discretization error due to truncation of the Fourier series at the mode N.

20 18 U. LANGER AND M. WOLFMAYR Theorem 5. Let us assume that y, p V 1. Then the discretization error due to truncation of the Fourier series can be estimated by (46) (y, p) (y N, p N ) P c 1 N (y, p) P1, where c is a constant depending only on the frequency ω. Proof. The P -norm is given by (y, p) (y N, p N ) 2 P = y y N 2 V + λ 1 p p N 2 V. Under the assumption that y V 1 = (H, 1 2 ) 1 ( ) Hper(Q,1 T ), we obtain the following estimate: y y N 2 V = T [ y 2 k 2 L 2 () + λ(ν y k, y k ) L2 () + ] λkω(σy k, y k ) L2 () T 2 T 2 k=n+1 [ 1 + kω kω y k 2 L 2 () + λ 1 + kω kω (ν y k, y k ) L 2 () + ] λ (kω)2 kω (σy k, y k ) L 2 () 1 [ (1 + kω) y kω k 2 L 2 () k=n+1 max k N+1 k=n+1 + λ(1 + kω)(ν y k, y k ) L2 () + λ(kω) 2 (σy k, y k ) L2 () 1 T [ (1 + kω) y (N + 1)ω 2 k 2 L 2 () k=n+1 + λ(1 + kω)(ν y k, y k ) L 2 () + ] λ(kω) 2 (σy k, y k ) L 2 () c 2 1 N y 2 V 1, where c = c (ω) = 1/ ω. Altogether we obtain for y, p V 1 the following estimate: (y, p) (y N, p N ) 2 P = y y N 2 V + λ 1 p p N 2 V c 2 1 N y 2 V 1 + λ 1 c 2 1 N p 2 V 1 = c 2 1 N (y, p) 2 P 1, hence (y, p) (y N, p N ) P c (1/ N) (y, p) P Discretization error in h. The discretization error between the multiharmonic approximation of the exact solution and its multiharmonic finite element approximation can be deduced to the discretization error between the unknown Fourier coefficients and their finite element approximations, see [2]. We have that (y N, p N ) (y Nh, p Nh ) 2 P = T (y, c p c ) (yh, c p c h) 2 P + T N (y 2 k, p k ) (y kh, p kh ) 2 P. The following theorem provides an estimate for discretization error between the unknown Fourier coefficients and their finite element approximations: Theorem 6. Under the assumption that (y k, p k ) (H 2 ()) 4 the discretization error for the Fourier coefficients can be estimated by (47) (y k, p k ) (y kh, p kh ) P c 1 c par (λ, k, ω, ν, σ, h) h (y k, p k ) H 2 (), ]

21 PARABOLIC CONTROL PROBLEM 19 where c 1 is a positive constant and c 2 par(λ, k, ω, ν, σ, h) = λνc 2 1,2+(1+kω λσ)c 2,2h 2 with constants c,2 and c 1,2 from the approximation theorem, and H 2 () is a weighted H 2 ()-seminorm defined by the relation (y k, p k ) 2 H 2 () = y k 2 H 2 () + λ 1 p k 2 H 2 (). Proof. See [2]. The following theorem provides the estimate for the complete discretization error in h: Theorem 7. Under the assumptions of Theorem 6, the discretization error in h can be estimated as follows: (48) (y N, p N ) (y Nh, p Nh ) P c 1 c par (λ, N, ω, ν, σ, h) h (y N, p N ) H 2, where c 2 par(λ, N, ω, ν, σ, h) = λνc 2 1,2 + (1 + Nω λσ)c 2,2h 2 with constants c,2 and c 1,2 from the approximation theorem and where the H 2 -seminorm is given by (y N, p N ) 2 H 2 = T (yc, p c ) 2 H 2 () + T 2 N (y k, p k ) 2 H 2 (), which is an H 2, ( )-seminorm defined in the Fourier space. The H 2 ()-seminorm for the Fourier coefficients is given in Theorem 6. Proof. The proof follows immediately from Theorem 6, see [2]. Remark 5. As it has been mentioned in [2], the convergence rate in (47), hence also in (48), reduces from h to h s with some s (, 1) if the Fourier coefficients of y and p only belong to H 1+s (). This result can be proven with the help of space interpolation theory, see [2, 8] Complete discretization error. We have to introduce new function spaces and corresponding norms such that we can estimate both, the P 1 -norm of the discretization error in Theorem 5 and the H 2 -seminorm of the discretization error in Theorem 7, by one norm. Definition 4. We define the following function spaces: (H, 1 2 ) 2 := {v H, 1 2 : v, 2 v H, 1 2 }, (H,1 per) 1 := {v H,1 per : v H,1 per}, X := (H, 1 2 ) 2 (H,1 per) 1, where we define the following norm in the space X: y 2 X = y 2 L 2 ( ) + 1/2 t y 2 L 2 () + t y 2 L 2 ( ) + y 2 L 2 ( ) + 1/2 t y 2 L 2 ( ) + t y 2 L 2 ( ) + 2 y 2 L 2 ( ) + 1/2 t 2 y 2 L 2 ( ) defined in the Fourier space according to Definition 2, i.e. y 2 X = T ( y c 2 L 2 () + yc 2 L 2 () + 2 y c 2 L 2 () ) + T [(1 + kω + (kω) 2 ) y 2 k 2 L 2 () + (1 + kω + (kω) 2 ) y k 2 L 2 () + (1 + kω) 2 y k 2 L 2 () ]. Moreover, we define the following vector-valued norm: (y, p) 2 X = y 2 X + λ 1 p 2 X.

22 2 U. LANGER AND M. WOLFMAYR Remark 6. The H 2 -seminorm of the space H 2, ( ) defined in Theorem 7 is bounded by the X-norm, i.e. y H 2 = T y c 2 H 2 () + T 2 y k 2 H 2 () y 2 X. The same bound is valid for the vector-valued versions of the H 2 -seminorm and the X-norm as well. The following lemma shows that the V 1 -norm and the P 1 -norm are also bounded by the X-norm and the vector-valued X-norm, respectively. Lemma 2. The P 1 -norm is bounded by the vector-valued X-norm, i.e. (49) (y, p) P1 c (y, p) X with the constant c 2 = max{1, λν, λσ}. Proof. We have that y 2 V 1 = T ( y c 2 L 2 () + λ(ν y, c y) c L2 ()) + T [(1 + kω) y 2 k 2 L 2 () + λ(1 + kω)(ν y k, y k ) L 2 () + λ(kω) 2 (σy k, y k ) L 2 ()] T ( y c 2 L 2 () + λν y c 2 L 2 () ) + T [(1 + kω + λσ(kω) 2 ) y 2 k 2 L 2 () + λν(1 + kω) y k 2 L 2 () ] max{1, λν, ( λσ} T ( y c 2 L 2 () + yc 2 L 2 () ) + T ) [(1 + kω + (kω) 2 ) y 2 k 2 L 2 () + (1 + kω) y k 2 L 2 () ] max{1, λν, ( λσ} T ( y c 2 L 2 () + yc 2 L 2 () + 2 y c 2 L 2 () ) + T [(1 + kω + (kω) 2 ) y 2 k 2 L 2 () + (1 + kω + (kω)2 ) y k 2 L 2 () + (1 + kω) 2 y k 2 L 2 () ] ) = c 2 y 2 X with c 2 = max{1, λν, λσ}. Now we can estimate the complete error arising from the multiharmonic finite element discretization. Theorem 8. Let us assume that y, p X. Then the complete discretization error arising from the multiharmonic finite element discretization can be estimated by ( 1 ) (5) (y, p) (y Nh, p Nh ) P C + c par (λ, N, ω, ν, σ, h) h (y, p) X, N where C = c max{c, c 1 } with c, c 1 and c from Theorem 5, Theorem 7 and Lemma 2, respectively, and c 2 par(λ, N, ω, ν, σ, h) = λνc 2 1,2+(1+Nω λσ)c 2,2h 2 with constants c,2 and c 1,2 from the approximation theorem.

23 PARABOLIC CONTROL PROBLEM 21 Proof. Applying triangle inequality and Theorems 5, 6 and 7 yields (y, p) (y Nh, p Nh ) P (y, p) (y N, p N ) P + (y N, p N ) (y Nh, p Nh ) P 1 c (y, p) P1 + c 1 c par (λ, N, ω, ν, σ, h) h (y N, p N ) H 2 N ( 1 ) max{c, c 1 } (y, p) P1 + c par (λ, N, ω, ν, σ, h) h (y N, p N ) H 2. N From Lemma 2 and Remark 6 follows that ( 1 (y, p) (y Nh, p Nh ) P c max{c, c 1 } (y, p) X N ) + c par (λ, N, ω, ν, σ, h) h (y N, p N ) X with the constant C = c max{c, c 1 }. ( 1 ) C + c par (λ, N, ω, ν, σ, h) h (y, p) X N Remark 7. From Theorem 8 we observe a relation between the discretization parameters N and h: in the case of having a low regularity assumption, we have to choose N = O(h 2 ) in order to get a good bound for the complete discretization error. 6. Numerical Results In this section, we present and discuss the results of our numerical experiments, where we study the numerical behavior of our multiharmonic finite element method. More precisely, we investigate the practical convergence behavior with respect to the space and time discretizations as well as the efficiency and the robustness of our preconditioned MINRES solver in different settings. We use the AMLI preconditioner proposed by Kraus in [24] for an inexact realization of our block-diagonal preconditioner in the MINRES method. First numerical results with this AMLI preconditioner in the context of multiharmonic parabolic optimal control problems were presented in [2], where the authors considered an optimal control problem with a given time-harmonic desired state. In this paper, we present new numerical results for the following more general settings: 1. the desired state is zero, and there is a given time-harmonic source term in the PDE constraints, 2. the desired state is periodic and analytic in time, 3. the desired state is analytic in time, but not time-periodic, 4. the desired state is a characteristic function in space and time, and 5. the desired state is a characteristic function in space and time, but in addition, there are jumps in ν and σ. We mention that the desired state is unreachable in Examples 3, 4 and 5. We consider our optimal control problem (1)-(3), where the computational domain = (, 1) (, 1) is uniformly decomposed into triangles, and standard continuous, piecewise linear finite elements are used for the discretization in space. The material coefficients are supposed to be piecewise constant on. In Examples 1, 2, 3 and 4, we set σ = ν = 1, but in Example 5, we consider jumping material coefficients. The desired states in Examples 2, 3, 4 and 5 are not time-harmonic. Therefore, we have to compute their Fourier coefficients for different modes k in order to expand the desired states into truncated Fourier series. The Examples 1 and 4 have been taken from [1], but with homogeneous Dirichlet boundary conditions instead of Robin conditions. We mention that, in all tables where the number of MINRES iterations is presented, the iteration was stopped after reducing the residual by a factor of 1 6. In each MINRES iteration step, we have used the AMLI preconditioner

24 22 U. LANGER AND M. WOLFMAYR according to [24] with 2 inner iterations. The computations for the figures of all examples were obtained on a grid. The results on grids of smaller mesh sizes were very similar. All computations were performed on a PC with Intel(R) Core(TM) i7-26 GHz. In the first example, the desired state y d is set, and the nontrivial timeharmonic source term f(x, t) = x 2 1(1 x 1 ) 2 x 2 2(1 x 2 ) 2 sin(2πt) is given. The frequency is set ω = 2π. Hence, we have the time period T = 1. This leads now to the following new Lagrange functional of our minimization problem: T ( L(y, u, p) := J (y, u) σ y t (ν y ) ) (51) u f p dx dt. Finally, the optimality conditions (16) yield the following reduced optimality system written in its classical formulation as σ(x) t y(x, t) (ν(x) y(x, t)) + λ 1 p(x, t) = f(x, t), (x, t), (52) y(x, t) =, (x, t) Σ T, y(x, ) = y(x, T ), x, σ(x) t p(x, t) (ν(x) p(x, t)) y(x, t) =, (x, t), p(x, t) =, (x, t) Σ T, p(x, T ) = p(x, ), x. The space-time variational formulation of (52) is obtained in the same way as for (18) in Subsection 2.2, and can be written in the following way: Given the source term f L 2 ( ), find y and p from H 1, 1 2 ( ) such that ( y v ν(x) p v + σ(x) 1/2 p 1/2 v ) dx dt =, t 1/2 t 1/2 (53) ( ν(x) y q + σ(x) 1/2 y 1/2 q ) t 1/2 t + 1/2 λ 1 p q dx dt = f v dx dt for all test functions v, q H 1, 1 2 ( ), where all functions are given in their Fourier series expansion in time according to Definition 2. We discretize the variational problem (53) by means of the multiharmonic finite element method. Since the source term is time-harmonic, we obtain the following linear system for k = 1 only: (54) M h K h ωm h,σ M h ωm h,σ K h K h ωm h,σ λ 1 M h ωm h,σ K h λ 1 M h y c y s p c p s = where f c = since the source term is only sine-wave-excited. As we see, only the right hand side of the optimality system and hence of the corresponding discretized system has changed. So, we can solve our problem in the same way as before and can use the same AMLI preconditioner as proposed in Section 4. Table 1 and Table 2 present the number of MINRES iterations and the computational times in seconds, respectively, varying the values of the parameter λ on grids of different mesh sizes. In Figure 1, we illustrate the harmonic source term and the finite element approximations of the control for different values of λ, more precisely, for λ {1, 1 2, 1 4, 1 6 }. Figure 2 presents the finite element approximations to the state for the same different values of λ. Both figures demonstrate that, for λ approaching, the control and the state approach the harmonic source term and the desired state, respectively. Compared to the methods used in [1], where the authors f c f s,