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1 SIAM J. NUMER. ANAL. Vol. 55, No. 6, pp c 2017 Society for Industrial and Applied Matematics EDGE ELEMENT METHOD FOR OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM WITH GAUSS LAW IRWIN YOUSEPT AND JUN ZOU Abstract. A novel edge element metod is proposed for te optimal control of te stationary Maxwell system wit a nonvanising carge density. Te proposed approac does not involve te usual saddle-point formulation and features a positive definite structure in te associated equality constraints, for wic optimal preconditioners are available in combination wit conjugate gradient iteration. Our main results include error estimates and strong convergence for bot te optimal edge element solution and te associated discrete Gauss laws. In particular, our analysis elps improve significantly te convergence rate establised by Ciarlet, Wu, and Zou [SIAM J. Numer. Anal., 52 (2014), pp ] for te edge element metod for te stationary Maxwell system. Numerical experiments are presented to verify te teoretical results. Key words. Maxwell equations, Gauss law, edge elements, optimality system, error estimates AMS subject classifications. 78A30, 78M50, 78M10 DOI /17M Introduction. Tis work sall examine an edge element approximation and te analysis of te following optimal control problem: (P) min 1 ɛ E E d 2 dx + κ ɛ u 2 dx, 2 2 Ω subject to te stationary Maxwell equations wit a nonvanising carge density curl (µ 1 curl E) = ɛu in Ω, (1) div (ɛe) = ρ in Ω, E n = 0 on Γ, and to te Gauss law for te applied current source (2) div (ɛu) = 0 in Ω. Te precise matematical assumptions on te data involved in (P) will be specified in section 2. Several matematical and numerical studies on electromagnetic optimal control problems can be found in te literature. However, tey were mainly focused on te cases were te stationary system (1) was replaced by te corresponding timedependent system [2, 15, 20, 21, 29] or te curl-curl-elliptic system [11, 26, 27]; namely, Received by te editors February 16, 2017; accepted for publication (in revised form) September 11, 2017; publised electronically November 14, ttp:// Funding: Te work of te first autor was supported by te German Researc Foundation (DFG) troug te SPP 1962 Non-Smoot and Complementarity-based Distributed Parameter Systems: Simulation and Hierarcical Optimization under project YO 159/2-1. Te work of te second autor was substantially supported by Hong Kong RGC General Researc Fund (project ) and NSFC/Hong Kong RGC Joint Researc Sceme (project N CUHK437/16). Fakultät für Matematik, Universität Duisburg-Essen, D Essen, Germany (irwin.yousept@ uni-due.de). Department of Matematics, Te Cinese University of Hong Kong, Satin, N.T., Hong Kong, Cina (zou@mat.cuk.edu.k) Ω

2 2788 IRWIN YOUSEPT AND JUN ZOU eiter a time derivative term E/ t or a zero-order term is added in te first equation of (1). In tese cases, te divergence law, i.e., te second equation in (1), can be automatically ensured at te discrete level from te first equation of (1) wen edge element metods are used for discretization. Tus, only two symmetric and positive definite systems (corresponding to (1) and its adjoint system) need to be solved in te discrete optimality conditions. However, te situation will be muc trickier and more difficult wen te stationary system (1) is considered instead of te corresponding time-dependent system or te curl-curl-elliptic system. Not muc as been studied for tis stationary optimal control problem except te recent work [28], were te optimal control (P), constrained wit te stationary state system (1) for ρ 0, ɛ 1, and a nonlinear magnetic permeability µ = µ(x, curl E ), was investigated bot matematically and numerically. In tis case, as most approximations do for te stationary state system (1), a Lagrange multiplier p is included in te left-and side of te first equation so tat (1) becomes a saddle-point system. A mixed finite element metod was proposed in [28] for tis saddle-point system using te lowest-order edge elements of Nédélec s first family and te continuous piecewise linear elements to approximate E and p, respectively. Te error estimates of te proposed finite element metod were also establised. Tese matematical and numerical results obtained in [28] are naturally valid for te stationary optimal control problem (P) wit te linear state system (1). We note tat two (resp., linearized) indefinite saddle-point systems (corresponding to (1) and its adjoint system) need to be solved at eac iteration (resp., eac inner iteration) wen an iterative metod is applied for solving te discretized version of (P) (resp., (P) constrained wit (1) wit a nonlinear magnetic permeability µ = µ(x, curl E )). Tis is an essential difference between te minimization (P) constrained wit te stationary state system (1) and its time-dependent version or te curl-curl-elliptic version. It is muc more difficult to solve te resulting indefinite saddle-point systems tan te similar symmetric and positive definite systems, for wic efficient and nearly optimal preconditioners are available suc as te multigrid and Hiptmair Xu preconditioners [8, 10] or te overlapping and nonoverlapping domain decomposition preconditioners [13, 22]. One of te most popular metods for solving suc discrete indefinite saddle-point systems is te preconditioned inexact Uzawa iterative metods, but tey converge in a reasonable rate only wen two efficient preconditioners are available for te curl-curl system and te corresponding Scur complement system (see [12, 13] and te references terein). But tis is usually quite difficult to realize in most applications. Tere is anoter fundamental issue tat needs our full attention wen we solve te optimal control problem (P) numerically. We see tat bot te continuous optimal solutions E and u satisfy te Gauss law (see (1) and (2)). It is pysically and matematically important weter te finite element metods used could guarantee te global strong convergence of te Gauss law for te discrete optimal solutions. Tis is still an open question for all existing finite element approximations of optimal control problems governed by bot stationary and nonstationary Maxwell systems. Tis work is mainly motivated by two numerical callenges we ave discussed above. In order to treat tese two numerical difficulties, a novel edge element metod was proposed recently for solving te stationary Maxwell equations (1) in [7] (for te case ρ = 0) and in [5] (for te general carge density). In contrast to most existing edge element scemes (see, e.g., [4, 6, 9, 18]), te new metod does not involve any saddlepoint structure. Instead, it requires only te resolution of a symmetric and positive definite system, wic can be solved by efficient and nearly optimal preconditioners, including [8, 10, 13, 22]. More importantly, te new edge element metod ensures

3 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2789 te optimal convergence rate [5, 7] and strong convergence of te Gauss law in some proper norm [5]. It is natural to ask weter te edge element metod [5, 7] wit all its advantages can be extended and transferred to te optimal control problem (P). Tis is exactly te main objective of te current work. On te basis of our earlier work in [5, 28], we terefore aim at developing an efficient finite element metod for te optimal control problem (P) witout a saddlepoint structure so tat te strong convergence of te Gauss law can be ensured for te discrete optimal solutions. We now describe our basic strategy to realize tis aim. In order to drop bot Gauss laws for te state E and te control u in (1) (2), we introduce two additional terms γɛe and γɛ χ wit a parameter γ > 0 in (1), were χ H 1 0 (Ω) is te unique solution of te variational equation: (3) (ɛ χ, ψ) L 2 (Ω) = (ρ, ψ) L2 (Ω) ψ H 1 0 (Ω). Tis leads to te following family of optimal control problems tat we consider: min 1 ɛ E E d 2 dx + κ ɛ u 2 dx, 2 (P γ Ω 2 Ω ) s.t. curl (µ 1 curl E) + γɛe = ɛ(u + γ χ) in Ω, E n = 0 on Γ. Hereafter, we discretize te state E and te control u in (P γ ) by te lowest-order edge elements of Nédélec s first family and consider γ as a function depending on te mes size. Based on tis concept, we propose te following finite element approximation: 1 min ɛ E E d 2 dx + κ ɛ u 2 dx, E,u N D 2 Ω 2 Ω (P ) s.t. (µ 1 curl E, curl v ) L 2 (Ω) + γ()(ɛe, v ) L 2 (Ω) = (ɛ(u + γ() χ ), v ) L 2 (Ω) v N D, were N D denotes te space of lowest-order edge elements of Nédélec s first family [19] wit vanising tangential traces. Furtermore, χ is an appropriate continuous piecewise linear approximation of χ. Te precise matematical formulation for (P ) will be presented in section 4. Te proposed finite element approac (P ) turns out to be very efficient, and tere are tree main reasons for tis, as we sall demonstrate later, were te second and tird ones present two important novel features in numerical solutions of te optimal control problem (P). First, te metod ensures strong convergence of (P ) towards (P) wit optimal convergence rate (Teorem 4.10). Second, it guarantees strong convergence of all Gauss laws involved, including te discrete optimal control, te discrete optimal state, and te discrete adjoint state (Teorem 4.7). More importantly, te equality constraint in (P ) features a positive definite structure; i.e., no saddle-point structure appears in (P ). Tis makes te resulting numerical metod muc more favorable tan te existing mixed finite element metods, especially wen te state Maxwell system (1) involves a nonlinear magnetic permeability µ = µ(x, curl E ) as considered in [28], were two linearized indefinite saddle-point systems need to be solved at eac inner iteration wen an iterative algoritm is applied for te optimal control problem. In addition, tere is anoter novel feature in our new formulation and metod, wic will be seen clearly in our subsequent numerical analysis: Te use of weigting coefficients ɛ and γɛ, respectively, in te objective functionals and te

4 2790 IRWIN YOUSEPT AND JUN ZOU state equations for (P γ ) and (P ) is crucial for te optimal convergence of te resulting finite element metod. Tis seems to be te first indication of te essential impact of te coefficients in te matematical and numerical studies of electromagnetic optimal control problems governed by Maxwell systems. Our strategy to prove error estimates for (P ) is based on te use of te solution operator of a discrete mixed variational problem (see section 4.2) in combination wit various optimal control and finite element tecniques. Here, for te proposed finite element metod (P ), we are able to prove te convergence rate of γ() + s for some exponent s (0.5, 1], depending on te regularity of te optimal solution to (P). In particular, tis result (see Corollary 4.12) significantly improves te recently obtained convergence rate of γ() + s for te edge element approximation of te stationary Maxwell system (1) wit a nonvanising carge density in [5]. We remark tat one drawback of our proposed metod lies in te stronger assumption on te desired electric field E d. We may notice tat te matematical and numerical analysis of te optimal control system (P) requires only E d L 2 (Ω) (see, e.g., [28]). But we need te additional assumption div (ɛe d ) = ρ L 2 (Ω) for our analyses in tis work. Tis condition appears to be reasonable from te pysical point of view, as it is in agreement wit te Gauss law about electricity (see Remark 2.3). Noneteless, te condition may not old if noisy data are allowed in te desired electric field E d. Te rest of tis paper is organized as follows. In next section, we introduce our notation and general assumptions for (P), including some preliminary results. Section 3 is devoted to te matematical analysis for (P) and (P γ ), including te strong convergence of (P γ ) towards (P) wit a reasonable convergence rate. In section 4, we analyze te finite element approximation (P ). Our main results include te strong convergence of te finite element solution wit optimal convergence rate and te strong convergence of te Gauss law in te discrete optimal state, te discrete optimal adjoint state, and te optimal discrete control. 2. Preliminaries. We start by introducing our notation and general assumptions for (P). Trougout tis work, unless it is specified explicitly, we sall use c to denote a generic positive constant, wic is independent of te mes size, te triangulation, and te quantities/fields of interest. For a given Hilbert space V, we use te notation V and (, ) V for a standard norm and a standard inner product in V. Te Euclidean norm in R 3 is denoted by. Furtermore, if V is continuously embedded in anoter normed function space Y, we write V Y. We use a bold typeface to indicate a tree-dimensional vector-valued function or a Hilbert space of tree-dimensional vector-valued functions. In our analysis, we mainly use te following Hilbert spaces: H(div) = { q L 2 (Ω) div q L 2 (Ω) }, H 0 (div) = { q H(div) q n = 0 on Γ }, H(div=0) = { q H(div) div q = 0 in Ω }, H(curl) = { q L 2 (Ω) curl q L 2 (Ω)}, H 0 (curl) = { q H(curl) q n = 0 on Γ }, were te div - and curl -operators as well as te tangential and normal traces are understood in te sense of distributions. Te state space associated wit (P) is given

5 by te Hilbert space OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2791 endowed wit te inner product V := { q H 0 (curl) ɛq H(div) }, (v, w) V := (v, w) H(curl) + (div (ɛv), div (ɛw)) L2 (Ω) v, w V and te norm V = (, ) 1/2 V. Furtermore, te control space associated wit (P) is given by te Hilbert space U := { u L 2 (Ω) ɛu H(div=0) }, endowed wit te inner product (, ) U = (, ) L 2 (Ω) and te norm U = (, ) 1/2 L 2 (Ω). Remark 2.1. It follows from te definition tat U = { u L 2 (Ω) ɛu H(div=0) } = { u L 2 (Ω) (ɛu, φ)l 2 (Ω) = 0 φ H 1 0 (Ω) }. Terefore, as proposed in [28], an ɛ-divergence-free control u U can be realized by including te variational equality (ɛu, φ) L 2 (Ω) = 0 φ H 1 0 (Ω) as an explicit control constraint of (P) in place of (2). But tis control constraint is naturally eliminated in (P γ ); see Remark 3.3. Assumption 2.2 (general assumptions for (P)). We assume tat Ω R 3 is a bounded domain wit a connected Lipscitz boundary Γ. Te electric permittivity ɛ : Ω R and te magnetic permeability µ : Ω R are of class L (Ω) and satisfy (4) 0 < µ µ(x) µ a.e. in Ω and 0 < ɛ ɛ(x) ɛ a.e. in Ω for some positive real constants µ < µ and ɛ < ɛ. Moreover, κ > 0 denotes te control cost constant, and te desired electric field E d L 2 (Ω) satisfies te Gauss law: (5) div (ɛe d ) = ρ in Ω (ɛe d, ψ) L 2 (Ω) = (ρ, ψ) L 2 (Ω) ψ H 1 0 (Ω), were ρ L 2 (Ω) is te carge density. Remark 2.3. In tis work, Ω represents a large oldall domain tat may contain different materials including conductors and inductors. We refer te reader to [24] for low-frequency electromagnetic optimal control problems wit multiply connected conductors. We note tat (5) arises from te Gauss law about electricity. As E d is te desired electric field, D d := ɛe d is ten te desired electric displacement field. According to te Gauss law about electricity, te divergence of te electric displacement field yields te free electric carge density, namely (5). We notice tat, since te boundary Γ is connected, tere exists a constant ĉ > 0, depending only on Ω, suc tat (6) E L 2 (Ω) ĉ ( ) curl E L 2 (Ω) + div (ɛe) L2 (Ω) E V. Te inequality (6) follows from a classical indirect argument by using te compactness of te embedding V L 2 (Ω) [25] and te fact tat {y V curl y = 0, div (ɛy) = 0} = {0},

6 2792 IRWIN YOUSEPT AND JUN ZOU wic olds due to te connectedness of Γ (see, e.g., [1]). Also, te Ladyzenskaya Babuška Brezzi (LBB) condition (ɛe, ψ) L (7) sup 2 (Ω) (ɛ ψ, ψ) L2 (Ω) c ψ 0 E H 0 (curl) E H(curl) ψ H 1 H(curl) 0 (Ω) ψ H0 1 (Ω) is satisfied wit a constant c > 0 depending only on ɛ and Ω. In fact, since H 1 0 (Ω) H 0 (curl) and curl 0, we may insert E = ψ in (7) to get te LBB condition. 3. Matematical analysis. We consider a mixed variational formulation for te stationary Maxwell equations (1): For a given u U, find E V suc tat { (µ 1 curl E, curl v) L 2 (Ω) = (ɛu, v) L 2 (Ω) v H 0 (curl), (8) (ɛe, ψ) L 2 (Ω) = (ρ, ψ) L2 (Ω) ψ H 1 0 (Ω). It is standard to verify tat, for every u U, te mixed variational formulation (8) admits a unique solution E V. Tis follows from a well-known teory for mixed variational problems (see [3]) togeter wit te Poincaré Friedrics-type inequality (6) and te LBB condition (7). Next, we introduce te solution operator associated wit (8) as G : U V, u E, tat assigns to every control u U te unique solution E V of te mixed variational formulation (8). Te solution operator G : U V is bounded and affine linear suc tat it is infinitely Frécet differentiable. Its Frécet derivative at z U in te direction u U is given by G (z)u = E z, were E z V is te solution of te following mixed variational equations: { (µ 1 curl E z, curl v) L 2 (Ω) = (ɛu, v) L 2 (Ω) v H 0 (curl), (9) (ɛe z, ψ) L 2 (Ω) = 0 ψ H 1 0 (Ω). Employing te solution operator, we may reformulate te optimal control problem (P) as a minimization problem in Hilbert spaces: (P) min f(u) := 1 ɛ G(u) E d 2 dx + κ ɛ u 2 dx. u U 2 2 Ω By classical arguments (see [16, 23]), te minimization problem (P) admits a unique solution u U, and its necessary and sufficient optimality condition is given by (10) f (u)u = 0 u U. Teorem 3.1. A control u U wit te associated electric field E V is te (unique) optimal solution of (P) if and only if tere exists a unique p V suc tat te triple (u, E, p) satisfies { (µ 1 curl E, curl v) L 2 (Ω) = (ɛu, v) L 2 (Ω) v H 0 (curl), (11a) (11b) (ɛe, ψ) L 2 (Ω) = (ρ, ψ) L2 (Ω) ψ H 1 0 (Ω), { (µ 1 curl p, curl v) L 2 (Ω) = (ɛ(e E d ), v) L 2 (Ω) v H 0 (curl), (ɛp, ψ) L 2 (Ω) = 0 Ω ψ H 1 0 (Ω), (11c) u = κ 1 p.

7 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2793 Proof. Te existence of a unique solution p V of te mixed variational problem (11b) follows from [3] along wit (5), (6), and (7). Inserting v = G (u)u wit u U in (11b) yields (12) (µ 1 curl p, curl (G (u)u)) L 2 (Ω) = (ɛ(e E d ), G (u)u) L 2 (Ω) u U. We also know tat G (u)u satisfies (9), wit z = u and E z = G (u)u, and ence inserting v = p in (9) gives (13) (µ 1 curl (G (u)u), curl p) L 2 (Ω) = (ɛu, p) L 2 (Ω) u U. From (11b), (12), and (13), we come to te conclusion tat f (u)u = (ɛ(e E d ), G (u)u) L 2 (Ω) + κ(ɛu, u) L 2 (Ω) = (ɛ(p + κu), u) L 2 (Ω) u U. Tus, te necessary and sufficient optimality condition (10) is noting but (14) (ɛ(p + κu), u) L 2 (Ω) = 0 u U. Now, te second variational equality in (11b) implies ɛp H(div=0). Tis regularity property implies tat p + κu U. Ten we can insert u = p + κu in (14) to obtain Tis completes te proof. (ɛ(p + κu), p + κu) L 2 (Ω) = 0 u = κ 1 p. In wat follows, we sall denote by u U te unique optimal solution of (P) wit te corresponding optimal electric field E V and te adjoint state p V U satisfying (11). Tanks to (11c), we can see tat te optimal control enjoys te regularity property (15) u V U Sensitivity analysis of (P γ ). Tis section is devoted to te sensitivity analysis of (P γ ), namely, to establis an error estimate depending on te parameter γ. First, we note tat te variational formulation for te associated state equation in (P γ ) is given by (16) (µ 1 curl E γ, curl v) L 2 (Ω) + γ(ɛe γ, v) L 2 (Ω) = (ɛ(u + γ χ), v) L 2 (Ω) v H 0(curl). By te Lax Milgram lemma, te variational equality (16) admits for every u L 2 (Ω) a unique solution E γ H 0 (curl). We denote te corresponding solution operator by G γ : L 2 (Ω) H 0 (curl), u E γ. Some elementary properties of tis operator are listed below for later use. Lemma 3.2. Te solution operator G γ : L 2 (Ω) H 0 (curl) satisfies G γ (0) = χ and div (ɛg γ (u)) = ρ for all u U and all γ > 0. Proof. Let γ > 0. Since curl 0, we can easily see tat (µ 1 curl ( χ), curl v) L 2 (Ω) + γ(ɛ χ, v) L 2 (Ω) = (ɛ(0 + γ χ), v) L 2 (Ω) v H 0 (curl), wic implies G γ (0) = χ by te definition of G γ. Now, for any u U, we insert v = ψ wit ψ H 1 0 (Ω) in (16) and use (3) to see tat E γ := G γ (u) satisfies (17) γ(ɛe γ, ψ) L 2 (Ω) = (ɛ(u + γ χ), ψ) L 2 (Ω) = γ(ρ, ψ) L2 (Ω) ψ H 1 0 (Ω).

8 2794 IRWIN YOUSEPT AND JUN ZOU Similarly to (P), we reformulate (P γ ) as a minimization problem in Hilbert spaces: (P γ ) min f γ (u) := 1 ɛ G γ (u) E d 2 dx + κ ɛ u 2 dx. u L 2 (Ω) 2 2 Ω Remark 3.3. We empasize tat te formulation (P γ ) removes te original divergence constraint on te control u as te control space of (P γ ) is now given by L 2 (Ω) instead of U as in (P). Noneteless, we will see later tat te optimal control of (P γ ) belongs to U. Similarly to (P), (P γ ) admits a unique optimal solution, wit its necessary and sufficient optimality conditions described as in te following teorem, wose proof is basically analogous to tat of Teorem 3.1. Teorem 3.4. Let γ > 0. A control u γ L 2 (Ω) wit te associated electric field E γ H 0 (curl) is te (unique) optimal solution of (P γ ) if and only if tere exists a unique p γ H 0 (curl) suc tat te triple (u γ, E γ, p γ ) satisfies (18a) (18b) (µ 1 curl E γ, curl v) L 2 (Ω) + γ(ɛe γ, v) L 2 (Ω) = (ɛ(u γ + γ χ), v) L 2 (Ω) Ω v H 0 (curl), (µ 1 curl p γ, curl v) L 2 (Ω) + γ(ɛp γ, v) L 2 (Ω) = (ɛ(e γ E d ), v) L 2 (Ω) (18c) u γ = κ 1 p γ. v H 0 (curl), An important consequence of te optimality system for (P γ ) is te following structural property for te optimal triple (u γ, E γ, p γ ) of (P γ ). Proposition 3.5. For every γ > 0, let (u γ, E γ, p γ ) L 2 (Ω) H 0 (curl) H 0 (curl) be te optimal triple of (P γ ) satisfying (18). Ten it olds tat (19) u γ V U, E γ V, div (ɛe γ ) = ρ, p γ V U. (20) Proof. For a fixed γ > 0, inserting v = ψ wit ψ H 1 0 (Ω) in (18a) yields γ(ɛe γ, ψ) L 2 (Ω) = (ɛ(u γ + γ χ), ψ) L 2 (Ω) = (ɛ( κ 1 p γ + γ χ), ψ) L 2 (Ω) = κ 1 (ɛp γ, ψ) L 2 (Ω) γ(ρ, ψ) L 2 (Ω) ψ H 1 0 (Ω), were we ave used (18c) and (3). Analogously, setting v = ψ wit ψ H 1 0 (Ω) in (18b) implies tat (21) γ(ɛp γ, ψ) L 2 (Ω) = (ɛ(e γ E d ), ψ) L 2 (Ω) From (20) and (21), it follows tat }{{} = (ɛe γ, ψ) L 2 (Ω) + (ρ, ψ) L 2 (Ω) (5) (γ 2 + κ 1 )(ɛp γ, ψ) L 2 (Ω) = 0 ψ H 1 0 (Ω). ψ H 1 0 (Ω). In oter words, div (ɛp γ ) = 0, so p γ V U. Ten it follows from (18c) tat u γ V U. Consequently, in view of Lemma 3.2, we obtain tat E γ V and div (ɛe γ ) = ρ. Tis completes te proof.

9 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2795 In wat follows, for every γ > 0, let (u γ, E γ, p γ ) (V U) V (V U) denote te optimal triple of (P γ ) satisfying (18). Teorem 3.6. Tere exists a constant c > 0, independent of γ, suc tat u γ u H(curl) + E γ E H(curl) + p γ p H(curl) cγ γ > 0. Proof. As u γ is te optimal solution of (P γ ), it follows tat f γ (u γ ) f γ 1 (0) }{{} = ɛ χ E d 2 dx γ > 0. 2 Ω Lemma 3.2 Tis implies te existence of a constant c > 0, independent of γ > 0, suc tat (22) E γ 2 L 2 (Ω) + uγ 2 L 2 (Ω) c γ > 0. Ten (22), along wit (18c), implies tat {u γ } γ>0, {E γ } γ>0, and {p γ } γ>0 are all bounded in L 2 (Ω). Setting v = p γ p in (18a) and (11a), respectively, ten subtracting te resulting equalities, we infer tat (23) (µ 1 curl (E γ E), curl (p γ p)) L 2 (Ω) + γ(ɛe γ, p γ p) L 2 (Ω) = (ɛ(u γ + γ χ u), p γ p) L 2 (Ω) }{{} = κ 1 ɛ 1/2 (p γ p) 2 L 2 (Ω) + γ(ɛ χ, pγ p) L 2 (Ω). (11c) (18c) Similarly, setting v = E γ E in (18b) and (11b), respectively, ten subtracting te resulting equations yields tat (24) (µ 1 curl (p γ p), curl (E γ E)) L 2 (Ω) + γ(ɛp γ, E γ E) L 2 (Ω) In view of (23) and (24), we obtain ɛ 1/2 (E γ E) 2 L 2 (Ω) + κ 1 ɛ 1/2 (p γ p) 2 L 2 (Ω) = γ ( (ɛp γ, E γ E) L 2 (Ω) + (ɛ( χ Eγ ), p γ ) p) L 2 (Ω) = ɛ 1/2 (E γ E) 2 L 2 (Ω) γ > 0. γ ( ɛ 1/2 p γ L 2 (Ω) + ɛ1/2 ( χ E γ ) L 2 (Ω))( ɛ 1/2 (E γ E) L 2 (Ω) + ɛ1/2 (p γ p) L 2 (Ω)). From te above estimate and te boundedness of {E γ } γ>0 and {p γ } γ>0 in L 2 (Ω), it follows tat E γ E L 2 (Ω) + p γ p L 2 (Ω) cγ γ > 0. Ten making use of (11c) and (33c), we ave (25) u γ u L 2 (Ω) + E γ E L 2 (Ω) + p γ p L 2 (Ω) cγ γ > 0. Now, inserting v = E γ E in (18a) and (11a), respectively, ten subtracting te resulting equations, we obtain tat (26) µ 1/2 curl (E γ E) 2 L 2 (Ω) = γ(ɛ( χ E γ ), E γ E) L 2 (Ω) + (ɛ(u γ u), E γ E) L 2 (Ω) γɛ χ E γ L 2 (Ω) E γ E L 2 (Ω) + ɛ u γ u L 2 (Ω) E γ E L 2 (Ω) γ > 0.

10 2796 IRWIN YOUSEPT AND JUN ZOU Analogously, we insert v = p γ p in (18b) and (11b) to obtain (µ 1 curl p γ, curl p γ p) L 2 (Ω) + γ(ɛp γ, p γ p) L 2 (Ω) = (ɛ(e γ E d ), p γ p) L 2 (Ω) and (µ 1 curl p, curl p γ p) L 2 (Ω) = (ɛ(e E d ), p γ p) L 2 (Ω). Ten, subtracting tese two identities yields (27) µ 1/2 curl (p γ p) 2 L 2 (Ω) = γ(ɛp γ, p γ p) L 2 (Ω) + (ɛ(e γ E), p γ p) L 2 (Ω) γɛ p γ L 2 (Ω) p γ p γ L 2 (Ω) + ɛ E γ E γ L 2 (Ω) p γ p γ L 2 (Ω). Now te desired estimate in Teorem 3.6 is a direct consequence of (25) (27). 4. Finite element metod. Tis section is devoted to te analysis of te finite element approximation (P ) we proposed in te introduction. From now on, te domain Ω R 3 is additionally assumed to be Lipscitz polyedral. We consider a family {T } >0 of triangulations of Ω consisting of tetraedral elements T suc tat Ω = T. T T For every element T T, we denote by T te diameter of T, by ρ T te diameter of te largest ball contained in T, and by te maximal diameter of all elements, i.e., := max{ T T T }. We assume {T } >0 is quasi-uniform, i.e., tere exist two positive constants ϱ and ϑ suc tat T ρ T ϱ and T ϑ T T, > 0. Let us denote te space of lowest-order edge elements of Nédélec s first family [19] wit vanising tangential traces and te space of continuous piecewise linear elements wit vanising traces by N D := { E H 0 (curl) E T = a T + b T x wit a T, b T R 3 T T }, Θ := { φ H 1 0 (Ω) φ T = a T x + b T wit a T R 3, b T R T T }. By te well-known discrete de Ram diagram (cf. [18, p. 150]), we know tat (28) Θ N D. In wat follows, we consider te parameter γ as a function of te mes size of te discretization; i.e., γ = γ(). Tis function is supposed to be bounded; i.e., tere exists a constant c > 0, independent of > 0, suc tat (29) 0 < γ() c > 0. Now we introduce te finite element solution χ Θ of (3): (30) (ɛ χ, ψ ) L 2 (Ω) = (ρ, ψ ) L 2 (Ω) ψ Θ. Ten we sall consider te following finite element approximation of (16).

11 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2797 For every given u L 2 (Ω), find E N D suc tat, for all v N D, it olds tat (31) (µ 1 curl E, curl v ) L 2 (Ω) + γ()(ɛe, v ) L 2 (Ω) = (ɛ(u + γ() χ ), v ) L 2 (Ω). We denote te (discrete) solution operator associated wit (31) by G : L 2 (Ω) N D, u E, tat assigns to every u L 2 (Ω) te unique solution E N D of (31). For later use, we introduce te subspace X (ɛ) of N D consisting of all discrete ɛ-divergence-free edge element functions: (32) X (ɛ) := { u N D (ɛu, ψ ) L 2 (Ω) = 0 ψ Θ }. Ten, making use of tis subspace, we can drive a discrete counterpart of Lemma 3.2. Lemma 4.1. For every > 0, te operator G : L 2 (Ω) N D satisfies G (0) = χ and (ɛg (u ), φ ) L 2 (Ω) = (ρ, φ ) L 2 (Ω) for all u X (ɛ) and φ Θ. Proof. For > 0, we can easily see by using curl 0 tat (µ 1 curl ( χ ), curl v ) L 2 (Ω) + γ()(ɛ χ, v ) L 2 (Ω) = γ()(ɛ χ, v ) L 2 (Ω) = (ɛ(0 + γ() χ ), v ) L 2 (Ω) v N D, wic implies G (0) = χ by using te definition of G and te fact tat Θ N D. Now, inserting v = ψ wit ψ Θ in (31), we see tat E := G (u ) for every u X (ɛ) satisfies γ()(ɛe, ψ ) L 2 (Ω) = (ɛ(u +γ() χ ), ψ ) L 2 (Ω) = γ()(ρ, ψ ) L 2 (Ω) ψ Θ, were te last equality olds due to u X (ɛ) and (30). Now, by introducing te objective functional f : L 2 (Ω) R, f (u) := 1 ɛ G (u) E d 2 dx + κ 2 2 we propose te finite element approximation for (P γ ) as follows: (P ) min u N D f (u ). Ω Ω ɛ u 2 dx, 4.1. Convergence analysis for (P ). For te convergence and error estimates of te finite element approximation (P ), we first present its necessary and sufficient optimality condition, wose proof is analogous to tat of Teorem 3.1. Teorem 4.2. Let > 0. A function u N D is te (unique) optimal solution of (P ) if and only if tere exists a unique p n N D suc tat te following olds for all v N D : (33a) (33b) (µ 1 curl E, curl v ) L 2 (Ω) + γ()(ɛe, v ) L 2 (Ω) = (ɛ(u + γ() χ ), v ) L 2 (Ω), (µ 1 curl p, curl v ) L 2 (Ω) + γ()(ɛp, v ) L 2 (Ω) = (ɛ(e E d ), v ) L 2 (Ω), (33c) u = κ 1 p.

12 2798 IRWIN YOUSEPT AND JUN ZOU Based on te optimality system (33) and Lemma 4.1, we obtain a discrete counterpart of Proposition 3.5. Tis result is essential to our convergence analysis. Proposition 4.3. For every > 0, let X (ɛ) be te space as defined in (32), and let u, E, p N D be te optimal triple of (P ) satisfying (33). Ten it olds tat (34) u, p X (ɛ) and (ɛe, ψ ) L 2 (Ω) = (ρ, ψ ) L 2 (Ω) ψ Θ, > 0. Proof. Tanks to (28), we may insert v = ψ wit ψ Θ in (33a) and use (33c) and (30) to obtain (35) γ()(ɛe, ψ ) L 2 (Ω) = (ɛ(u + γ() χ ), ψ ) L 2 (Ω) = κ 1 (ɛp, ψ ) L 2 (Ω) + γ()(ɛ χ, ψ ) L 2 (Ω) = κ 1 (ɛp, ψ ) L 2 (Ω) γ()(ρ, ψ ) L2 (Ω) ψ Θ. Similarly, inserting v = ψ wit ψ Θ in (33b) yields (36) γ()(ɛp, ψ ) L 2 (Ω) = (ɛ(e E d ), ψ ) L 2 (Ω) = (ɛe, ψ ) L 2 (Ω) + (ρ, ψ ) L 2 (Ω) ψ Θ, were we ave used (5). Ten we infer from (35) and (36) tat γ() 2 (ɛp, ψ ) L 2 (Ω) = γ() ( ) (ɛe, ψ ) L 2 (Ω) + (ρ, ψ ) L2 (Ω) = κ 1 (ɛp, ψ ) L 2 (Ω) ψ Θ, from wic it follows tat (γ() 2 + κ 1 )(ɛp, ψ ) L 2 (Ω) = 0 for all ψ Θ. Tus, we come to te desired conclusion tat p X (ɛ) = }{{} (33c) u X (ɛ) }{{} = (ɛe, ψ ) L 2 (Ω) = (ρ, ψ ) L 2 (Ω) ψ Θ. Lemma 4.1 Te upcoming lemma states te discrete compactness property for X (ɛ). Te discrete compactness property for Nédélec s edge elements in te case ɛ 1 goes back to Kikuci [14]. Lemma 4.4. Let {z } >0 be a uniformly bounded sequence in H 0 (curl) satisfying z X (ɛ) for all > 0. Ten, tere exists a subsequence {z n } n=1 {z } >0 wit n 0 as n suc tat z n z strongly in L 2 (Ω) as n, curl z n curl z weakly in L 2 (Ω) as n for some z H 0 (curl) U, i.e., div (ɛz) = 0 in Ω. Proof. Te assertion is well known (see, e.g., [18]). We provide te proof only for te convenience of te reader. In view of te discrete Helmoltz decomposition, for every > 0, tere exists a unique pair (y 1, θ1 ) X(1) Θ suc tat (37) z = y 1 + θ 1.

13 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2799 Due to te uniform boundedness {z } >0 H 0 (curl), te sequence {y 1 } >0 is uniformly bounded in H 0 (curl). Tus, employing te discrete compactness property [14] for ɛ 1, we find a subsequence {y 1 n } n=1 {y 1 } >0 wit n 0 as n suc tat (38) y 1 n y 1 strongly in L 2 (Ω) as n, curl y 1 n curl y 1 weakly in L 2 (Ω) as n for some y 1 H 0 (curl) H(div=0). Now, making use of te classical Helmoltz decomposition, tere exists a unique pair (y ɛ, θ ɛ ) H 0 (curl) U H 1 0 (Ω) suc tat (39) y 1 = y ɛ + θ ɛ. We now sow tat z n y ɛ strongly in L 2 (Ω) as n. Since y ɛ U and z n X (ɛ) n olds for all n N, (40) (ɛ(z n y ɛ ), φ n ) L 2 (Ω) = 0 φ n Θ n, n N. From (37), (39), and (40), we obtain tat (ɛ(z n y ɛ ), z n y ɛ ) L 2 (Ω) = (ɛ(z n y ɛ ), y 1 n y 1 + θ ɛ φ n ) L 2 (Ω) φ n Θ n, n N, and so ɛ z n y ɛ L 2 (Ω) ɛ y 1 n y 1 L 2 (Ω) + ɛ θ ɛ φ n L 2 (Ω) φ n Θ n, n N. Now employing (38) and te fact tat {Θ n } n=1 is dense in H 1 0 (Ω), te above inequality implies te strong convergence z n y ɛ in L 2 (Ω) as n. In wat follows, for every > 0, we sall denote by (u, E, p ) X (ɛ) N D X (ɛ) te optimal triple of (P ) satisfying (33). Let us now prove te strong convergence of (u, E, p ) to (u, E, p) as 0 in te following teorem. Teorem 4.5. Suppose tat lim 0 γ() = 0. Ten, lim 0 u u H(curl) = lim 0 E E H(curl) = lim 0 p p H(curl) = 0. Proof. For every > 0, te fact tat u is te unique solution of (P ) yields f (u ) f (0) }{{} = 1 2 ɛ1/2 ( χ E d ) 2 L 2 (Ω) > 0. Lemma 4.1 Terefore, in view of (30), tere exists a constant c > 0, independent of, suc tat (41) E L 2 (Ω) + u L 2 (Ω) c > 0 }{{} = (33c) p L 2 (Ω) c > 0. Now, setting v = E in (33a) and v = p in (33b) and ten employing (41) and (29), we obtain (42) curl E L 2 (Ω) c and curl p L 2 (Ω) c > 0.

14 2800 IRWIN YOUSEPT AND JUN ZOU From (41) and (42), we conclude tat te sequences {u } >0, {E } >0, and{p } >0 are all uniformly bounded in H 0 (curl). Terefore, tere exists a subsequence {(u n, E n, p n )} n=1 {(u, E, p )} >0 wit n 0 as n suc tat u n ũ weakly in H 0 (curl) as n, (43) E n Ẽ weakly in H 0(curl) as n, p n p weakly in H 0 (curl) as n for some ũ, Ẽ, p H 0(curl). From (33c), we know u n = κ 1 p n for all n N. Tus, (43) implies tat (44) ũ = κ 1 p. Now, we denote by I : C0 (Ω) Θ te nodal interpolation operator corresponding to te finite element space Θ. By virtue of Proposition 4.3, it olds for every ψ C0 (Ω) tat (45) (ɛu n, I n ψ) L 2 (Ω) = 0 n N, (ɛp n, I n ψ) L 2 (Ω) = 0 n N, (ɛe n, I n ψ) L 2 (Ω) = (ρ, I ψ) L2 (Ω) n N. Ten, passing to te limit n in (45), we obtain from (43) tat (ɛũ, ψ) L 2 (Ω) = (ɛ p, ψ) L 2 (Ω) = 0 and (ɛẽ, ψ) L 2 (Ω) = (ρ, ψ) L2 (Ω) ψ C 0 (Ω). Consequently, since C0 (Ω) H0 1 (Ω) is dense, we come to te conclusion tat (ɛũ, ψ) L 2 (Ω) = 0 ψ H0 1 (Ω), (46) (ɛ p, ψ) L 2 (Ω) = 0 ψ H0 1 (Ω), (ɛẽ, ψ) L 2 (Ω) = (ρ, ψ) L2 (Ω) ψ H0 1 (Ω). Next, let N : C0 (Ω) N D denote te curl-conforming Nédélec interpolation operator corresponding to te finite element space N D. According to (33a), we ave for every v C0 (Ω) tat (47) (µ 1 curl E n, curl N n v) L 2 (Ω) + γ( n )(ɛe n, N n v) L 2 (Ω) = (ɛ(u n + γ( n ) χ n ), N n v) L 2 (Ω) n N. Passing to te limit n in (47), we obtain from (43) and lim n γ( n ) = 0 tat Terefore, as C 0 (µ 1 curl Ẽ, curl v) L 2 (Ω) = (ɛũ, v) L 2 (Ω) v C 0 (Ω). (Ω) H 0(curl) is dense, it follows tat (48) (µ 1 curl Ẽ, curl v) L 2 (Ω) = (ɛũ, v) L 2 (Ω) v H 0 (curl). Analogously, we deduce from (33b), (43), and lim n γ( n ) = 0 tat (49) (µ 1 curl p, curl v) L 2 (Ω) = (ɛ(ẽ E d), v) L 2 (Ω) v H 0 (curl). We can see from (44), (46), and (48) (49) tat te weak limit (ũ, Ẽ, p) satisfies te necessary and sufficient optimality condition for (P), and consequently (ũ, Ẽ, p) = (u, E, p).

15 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2801 In particular, te weak limit is independent of te subsequence {(u n, E n, p n } n=1, and consequently (43) olds for te wole sequence, i.e., u u weakly in H 0 (curl) as 0, (50) E E weakly in H 0 (curl) as 0, p p weakly in H 0 (curl) as 0. Now, making use of Lemma 4.4, we obtain from Proposition 4.3 and (50) tat { u u strongly in L 2 (Ω) as 0, (51) p p strongly in L 2 (Ω) as 0. Setting v = v = p in (33b) and (11b) yields (µ 1 curl (p p), curl p ) L 2 (Ω) + γ()(ɛp, p ) L 2 (Ω) = (ɛ(e E), p ) L 2 (Ω), from wic it follows tat µ 1/2 curl (p p) 2 L 2 (Ω) = γ()(ɛp, p ) L 2 (Ω) + (ɛ(e E), p ) L 2 (Ω) (µ 1 curl (p p), curl p) L 2 (Ω). Ten, passing to te limit 0, (50), (51), and lim 0 γ() = 0 imply lim curl (p p) L 0 2 (Ω) = 0. Togeter wit (51), tis strong convergence yields (52) lim p p H(curl) = 0 0 }{{} = (33c) lim u u H(curl) = 0. 0 It remains now to prove te strong convergence of {E } >0 in H 0 (curl). First, we verify te strong convergence in L 2 (Ω) by inserting v = v = E in (33b) and (11b): (µ 1 curl (p p), curl E ) L 2 (Ω) + γ()(ɛp, E ) L 2 (Ω) = (ɛ(e E), E ) L 2 (Ω), from wic it follows tat (53) ɛ 1/2 (E E) 2 L 2 (Ω) =(µ 1 curl (p p), curl E ) L 2 (Ω) + γ()(ɛp, E ) L 2 (Ω) (ɛ(e E), E) L 2 (Ω). Ten, passing to te limit 0 in (53), we obtain from (50), (52), and lim 0 γ() = 0 tat (54) lim 0 E E L 2 (Ω) = 0. Similarly, by setting v = v = E in (33a) and (11a), we deduce from (50), (52), and lim 0 γ() = 0 tat (55) lim 0 curl (E E) L 2 (Ω) = 0. From (54) (55), we come to te conclusion tat lim 0 E E H(curl) = 0.

16 2802 IRWIN YOUSEPT AND JUN ZOU 4.2. Error estimates. We first sow tat te newly proposed finite element approximation (P ) ensures te global strong convergence of all tree Gauss laws for te discrete optimal control, state, and adjoint state u, E, and p, wic satisfy te optimality system (33). We recall tat, using te fact tat Ω is Lipscitz polyedral, tere is a constant δ (0.5, 1] suc tat [1] (56) H 0 (curl) H(div) H δ (Ω) and H(curl) H 0 (div) H δ (Ω). Te results in te following lemma were verified in [5, Lemma 3.9]. Lemma 4.6. Suppose tat ɛ W 1, (Ω) and s (0.5, 1]. Ten, tere exists a constant c > 0, independent of and z, suc tat div (ɛz ) H s (Ω) c s+δ 1 curl z L 2 (Ω) for all > 0 and all z X (ɛ). Moreover, te solution χ Θ of (30) satisfies div (ɛ χ ) ρ H s (Ω) c s+δ 1 ρ H δ 1 (Ω) > 0. Teorem 4.7. Suppose tat ɛ W 1, (Ω), and s (0.5, 1]. Ten, tere exists a positive constant c, independent of, u, E, and p, suc tat for all > 0, div(ɛu ) H s (Ω) + div(ɛp ) H s (Ω) + div(ɛe ) ρ H s (Ω) c s+δ 1. Proof. From Proposition 4.3, we know tat u, p X (ɛ) for all > 0. Terefore, Lemma 4.6 togeter wit te uniform boundedness of {u } >0 and {p } >0 in H 0 (curl) (see Teorem 4.5) implies (57) div (ɛu ) H s (Ω) + div (ɛp ) H s (Ω) c s+δ 1 > 0. Making use again of Proposition 4.3 along wit (30), we ave tat (ɛe, ψ ) L 2 (Ω) = (ρ, ψ ) L2 (Ω) = (ɛ χ, ψ ) L 2 (Ω) ψ Θ, from wic it follows tat (ɛ(e χ ), ψ ) L 2 (Ω) = 0 ψ Θ = E χ X (ɛ) > 0. Ten using Lemma 4.6 we can derive (58) div (ɛe ) ρ H s (Ω) div (ɛ(e χ )) H s (Ω) + div (ɛ χ ) ρ H s (Ω) c s+δ 1 ( curl E L 2 (Ω) + ρ H δ 1 (Ω)) > 0. Terefore, since {E } >0 is uniformly bounded in H 0 (curl), te desired assertion follows from (57) (58). As our main goal, we will derive next te error estimates for te optimal control, state, and adjoint state of te proposed edge element metod (P ). To do so, we introduce te following discrete mixed variational problem. For a given E H 0 (curl), find te solution E = Φ (E) N D to { (µ 1 curl E, curl v ) L 2 (Ω) = (µ 1 curl E, curl v ) L 2 (Ω) v N D, (59) (ɛe, ψ ) L 2 (Ω) = (ɛe, ψ ) ψ Θ.

17 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2803 It is standard to verify tat, for every E H 0 (curl), te mixed discrete variational problem (59) admits a unique solution Φ E := E N D for all > 0, satisfying ( ) (60) Φ E E H(curl) c inf v E H(curl) E H 0(curl). v N D Tis follows again from a well-known teory for mixed variational problems (see, e.g., [18, Teorem 2.45]) by utilizing (6) (7), te discrete Poincaré Friedrics-type inequality [9, Teorem 4.7], (61) E L 2 (Ω) c curl E L 2 (Ω) and te discrete LBB condition, E X (ɛ), > 0, (ɛe, ψ ) L2(Ω) (62) sup (ɛ ψ, ψ ) L2 (Ω) c ψ 0 E N D E H(curl) ψ H 1 H(curl) 0 (Ω) ψ Θ, wit a constant c > 0 depending only on ɛ and Ω. Notice tat (62) olds due to te inclusion Θ N D. Now, making use of te operator Φ : H 0 (curl) N D, we obtain te following important identity for our subsequent analysis. (63) Lemma 4.8. It olds for all > 0 tat µ 1/2 curl (E Φ E) 2 L 2 (Ω) + κ 1 µ 1/2 curl (p Φ p) 2 L 2 (Ω) = γ() [ (ɛ( χ E ), E Φ E) L 2 (Ω) + κ 1 ] (ɛp, Φ p p ) L 2 (Ω) + κ 1 (ɛ(e Φ E), p Φ p) L 2 (Ω) + κ 1 (ɛ(φ E E), p Φ p) L 2 (Ω). Proof. In view of te state equations (18a) and (33a), we ave tat (µ 1 curl (E E), curl v ) L 2 (Ω) + γ()(ɛe, v ) L 2 (Ω) = (ɛ(u u + γ() χ ), v ) L 2 (Ω) v N D. Making use of te operator Φ and setting v = E Φ E in (63), we obtain µ 1/2 curl (E Φ E) 2 L 2 (Ω) + γ()(ɛe, E Φ E) L 2 (Ω) = (ɛ(u u + γ() χ ), E Φ E) L 2 (Ω) }{{} = κ 1 (ɛ(p p), E Φ E) L 2 (Ω) + γ()(ɛ χ, E Φ E) L 2 (Ω) (18c)&(33c) = κ 1 (ɛ(p Φ p), E Φ E) L 2 (Ω) + κ 1 (ɛ(p Φ p), E Φ E) L 2 (Ω) wic implies (64) + γ()(ɛ χ, E Φ E) L 2 (Ω), µ 1/2 curl (E Φ E) 2 L 2 (Ω) + κ 1 (ɛ(p Φ p), E Φ E) L 2 (Ω) = γ()(ɛ( χ E ), E Φ E) L 2 (Ω) + κ 1 (ɛ(p Φ p), E Φ E) L 2 (Ω). Similarly, we deduce from te adjoint equations (18b) and (33b) tat (65) (µ 1 curl (p p), curl v ) L 2 (Ω) + γ()(ɛp, v ) L 2 (Ω) = (ɛ(e E), v ) L 2 (Ω) v N D.

18 2804 IRWIN YOUSEPT AND JUN ZOU Ten, making use of te operator Φ and setting v = p Φ p in (65), we derive wic implies (66) µ 1/2 curl (p Φ p) 2 L 2 (Ω) + γ()(ɛp, p Φ p) L 2 (Ω) = (ɛ(e E), p Φ p) L 2 (Ω) = (ɛ(e Φ E), p Φ p) L 2 (Ω) + (ɛ(φ E E), p Φ p) L 2 (Ω), (ɛ(e Φ E), p Φ p) L 2 (Ω) = µ 1/2 curl (p Φ p) 2 L 2 (Ω) + γ()(ɛp, p Φ p) L 2 (Ω) (ɛ(φ E E), p Φ p) L 2 (Ω). Applying (66) to (64), we come to te desired identity: (67) µ 1/2 curl (E Φ E) 2 L 2 (Ω) + κ 1 µ 1/2 curl (p Φ p) 2 L 2 (Ω) = γ()(ɛ( χ E ), E Φ E) L 2 (Ω) + κ 1 (ɛ(p Φ p), E Φ E) L 2 (Ω) + κ 1 γ()(ɛp, Φ p p ) L 2 (Ω) + κ 1 (ɛ(φ E E), p Φ p) L 2 (Ω). We now recall a classical error estimate for te curl-conforming Nédélec interpolant N in te space H s (curl) := {E H s (Ω) curl E H s (Ω)} [6]. Lemma 4.9. For s (1/2, 1], tere exists a constant c > 0, independent of and E, suc tat for all > 0, (68) E N E H(curl) c s E H s (curl) We are now ready to establis our main result. E H s (curl). Teorem Suppose tat E, p H s (curl) for some s (0.5, 1]. Ten, tere exists a constant c > 0, independent of, u, E, and p, suc tat for all > 0. E E H(curl) + p p H(curl) + u u H(curl) c(γ() + s ) Proof. In view of te regularity assumption E, p H s (curl) wit s (0.5, 1] along wit (60) and (68), tere is a constant c > 0, independent of, suc tat (69) Φ E E H(curl) + Φ p p H(curl) c s > 0. On te oter and, according to Lemma 4.8, we ave te following estimate: µ 1 curl (E Φ E) 2 L 2 (Ω) + κ 1 µ 1 curl (p Φ p) 2 L 2 (Ω) ( γ() ( ɛ( χ E ) L 2 (Ω) + κ 1 ɛp L (Ω)) 2 + κ 1 ɛ(p Φ p) L 2 (Ω) )( ) + κ 1 ɛ(e Φ E) L 2 (Ω) E Φ E L 2 (Ω) + p Φ p L 2 (Ω) > 0. Ten applying (69) to te above estimate yields (70) ( curl (E Φ E) L 2 (Ω) + curl (p Φ p) L 2 (Ω)) 2 c(γ() + s )( E Φ E L 2 (Ω) + p Φ p L 2 (Ω)) > 0.

19 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM 2805 Using te definition of Φ and Proposition 4.3, we ave for every > 0 tat (ɛ(e Φ E), ψ ) L 2 (Ω) = (ɛ(e E), ψ ) L 2 (Ω) = 0 ψ Θ, (ɛ(p Φ p), ψ ) L 2 (Ω) = (ɛ(p p), ψ ) L 2 (Ω) = 0 ψ Θ. In oter words, it olds tat E Φ E X (ɛ) and p Φ p X (ɛ) > 0, so it follows from te discrete Poincaré Friedrics-type inequality (61) tat (71) E Φ E L 2 (Ω) + p Φ p L 2 (Ω) c ( curl (E Φ E) L 2 (Ω) ) + curl (p Φ p) L 2 (Ω) > 0. Applying (71) to (70), we deduce tat (72) curl (E Φ E) L 2 (Ω) + curl (p Φ p) L 2 (Ω) c(γ() + s ) > 0. Combining te estimates (71) (72) along wit (69), we finally obtain te estimate (73) E E H(curl) + p p H(curl) c(γ() + s ) > 0. Now te desired estimate follows from tis estimate above and te optimality conditions (11c) and (33c). Remark We can easily observe tat Teorem 4.10 ensures te optimal convergence rate for our proposed finite element optimal control metod (P ) if we take γ = O(). Note tat our analysis can elp improve te error estimate in [5]. In fact, by making use of te operator Φ, we are able to significantly improve te convergence rate of γ() + s acieved in [5] for te edge element approximation of te stationary Maxwell system (1) wit a nonvanising carge density. Our improved result is provided in te following corollary, wose proof is analogous to tat of Teorem Corollary Let f H(div=0) and z H 0 (curl) denote te unique solution of { (µ 1 curl z, curl v) L 2 (Ω) = (f, v) L 2 (Ω) v H 0 (curl), (74) (ɛz, ψ) L 2 (Ω) = (ρ, ψ) L2 (Ω) ψ H 1 0 (Ω). Furtermore, for every > 0, let z N D denote te unique solution of (75) (µ 1 curl z, curl v ) L 2 (Ω) + γ()(ɛz, v ) L 2 (Ω) = (f + γ()ɛ χ, v ) L 2 (Ω) were χ Θ is te solution of (76) (ɛ χ, ψ ) L 2 (Ω) = (ρ, ψ ) L 2 (Ω) ψ Θ. v N D, Ten, if z H s (curl) for some s (0.5, 1], tere exists a constant c > 0, independent of, z, and z, suc tat z z H(curl) c(γ() + s z H s (curl)) > 0.

20 2806 IRWIN YOUSEPT AND JUN ZOU Proof. Tanks to te regularity assumption z H s (curl), we obtain from (60) and (68) tat (77) z Φ z H(curl) c s z H s (curl) > 0. Now, making use of te operator Φ (see (59) for its definition), we infer tat (µ 1 curl (z Φ z), curl v ) L 2 (Ω) = (µ 1 curl (z z), curl v ) L 2 (Ω) }{{} = γ()(ɛz, v ) L 2 (Ω) + (f + ɛγ() χ, v ) L 2 (Ω) (f, v ) L 2 (Ω) (75)&(74) = γ()(ɛ( χ z ), v ) L 2 (Ω) v N D. Tus, inserting v = z Φ z N D, we obtain tat (78) µ 1/2 curl (z Φ z) 2 L 2 (Ω) = γ()(ɛ( χ z ), z Φ z) L 2 (Ω) γ() ɛ( χ z ) L 2 (Ω) z Φ z L 2 (Ω) cγ() z Φ z L 2 (Ω) > 0. On te oter and, by te definition of Φ (see (59)) and (74) (76) we infer tat (ɛ(z Φ z), ψ ) = (ɛ(z z), ψ ) = (ɛz, ψ ) + (ρ, ψ ) = (ɛ χ, ψ ) + (ρ, ψ ) = 0 ψ Θ, > 0. Consequently, we ave z Φ z X (ɛ) for all > 0, so we may apply te discrete Poincaré Friedrics-type inequality (61) to (78) to deduce tat µ 1 curl (z Φ z) L 2 (Ω) cγ() > 0. Ten, tis inequality togeter wit (61) implies (79) z Φ z H(curl) cγ() > 0. Finally, we obtain from (77) and (79) tat Tis completes te proof. z z H(curl) z Φ z H(curl) + z Φ z H(curl) c(γ() + s z H s (curl)) > 0. Remark We know from Teorem 4.10 (cf. Corollary 4.12) tat te coice γ() = yields te desired optimal error estimate of order O( s ), were te index s (0.5, 1] is determined by te regularity of te true solution. We recall tat te results in [5] require te coice γ() = 2 for te same optimal estimate O( s ). However, if is sufficiently small, ten te coice γ() = 2 is muc smaller tan γ() = and increases te numerical effort considerably because te conditioning of te edge element system (31) is muc worse. For tis reason, we suggest coosing γ() = for te numerical solution of (P ) to acieve te optimal error estimate wit a reduced computational effort.

21 OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM Numerical experiments. We present two numerical examples serving as a numerical illustration of Teorems 4.5 and Example 1 wit a smoot optimal solution. As te first example, we consider te model optimal control problem (P) tat as an analytical and smoot optimal solution, wit te computational domain Ω = (0, 1) 3, te parameters µ = ɛ = 1, ρ = 0, and κ = 1, and te desired state E d given by sin(πx 2 ) sin(πx 3 ) E d (x) = (4π 4 + 1) 0 0. Ten by straigtforward computations we can verify tat te tree functions sin(πx 2 ) sin(πx 3 ) sin(πx 2 ) sin(πx 3 ) E(x) = 0, u(x) = 2π 2 0, 0 sin(πx 2 ) sin(πx 3 ) p(x) = 2π satisfy te sufficient and necessary optimality system (11). Tus, te optimal solution of (P) is given by u. For all te examples in tis section, we ave solved te finite element approximation (P ) using te open source software FEniCS [17]. Te computational domain Ω was triangulated wit a regular mes of mes size, and te optimality system (33) was solved by MUMPS (MUltifrontal Massively Parallel sparse direct Solver). As pointed out in Remarks 4.11 and 4.13, in order to guarantee te optimal convergence rate in te finite element solution, we coose γ() =. Furtermore, we employ te following quantity to compute te approximate order of convergence: EOC = log u 1 u H(curl) log u 2 u H(curl) log 1 log 2 for two consecutive mes sizes 1 and 2. Table 1 displays te H(curl)-norm error between te analytical solution u and te finite element solution u for different mes sizes. As we can see from te table, te finite element solution u converges to te analytical solution u as decreases. Moreover, by Teorem 4.10 we know a convergence rate of s = 1 sould be obtained due to te nice regularity properties u, E, p H 1 (curl) for tis example. Tis teoretical prediction is confirmed by our numerical results, as we see EOC approximates s Example 2 wit a nonsmoot optimal solution. In tis example, we coose te nonconvex polyedral computational domain (80) Ω = { (0, 1/4) (0, 1/2) (0, 1) } \ { [1/8, 1/4] [1/8, 1/2] [0, 1] } and te parameters µ = ɛ = 1, ρ = 0, and κ = 1. For convenience, we now include an additional sift control in our objective functional: 1 (P) min G(u) E d 2 dx + 1 u u d 2 dx. u U 2 2 Ω Ω 0

22 2808 IRWIN YOUSEPT AND JUN ZOU Table 1 Convergence istory. / 2 u u H(curl) EOC Here, te desired state and te sift control are set to be E d = G(u d ) and u d = G(f), wit f = 10 3 (1, 1, 1) T. We note tat, since µ ɛ 1 and ρ 0, te desired state and te sift control enjoy te regularity property E d, u d H δ (curl), wit δ as in (56). As Ω is a nonconvex Lipscitz polyedron, tis exponent is strictly less tan one, δ (0.5, 1). We also point out tat te analytical solutions for E d = G(u d ) and u d = G(f) are unknown. For our numerical experiment, we approximate tem by teir finite element approximations wit a very fine mes size = By our specific construction, te optimal solution of (P) is exactly given by u = u d, and all our results in tis work can be naturally extended to (P) in te presence of te sift control u d. Table 2 displays te H(curl)-norm error between te exact solution u and our finite element solution u wit γ() =. As te mes size decreases, we observe tat te optimal solution approaces te exact one. See Figure 1 for te computed optimal electric field wit different meses. Furtermore, by Teorem 4.10 we know we can only expect a convergence rate δ (0.5, 1), as te computational domain (80) features a nonconvex structure in tis example. Tis teoretical prediction is also reasonably confirmed by our numerical results wit δ 0.7. Table 2 Convergence istory. / 2 u u H(curl) EOC

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