Bowen Li and Jun Zou
Transcript of Bowen Li and Jun Zou
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ESAIM: M2AN 55 (2021) 2013β2044 ESAIM: Mathematical Modelling and Numerical Analysishttps://doi.org/10.1051/m2an/2021046 www.esaim-m2an.org
AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE FORAN ELECTROMAGNETIC CONSTRAINED OPTIMAL CONTROL PROBLEM
Bowen Li1 and Jun Zou2,*
Abstract. In this work, an adaptive edge element method is developed for an H(curl)-elliptic con-strained optimal control problem. We use the lowest-order Nedelecβs edge elements of first family andthe piecewise (element-wise) constant functions to approximate the state and the control, respectively,and propose a new adaptive algorithm with error estimators involving both residual-type error esti-mators and lower-order data oscillations. By using a local regular decomposition for H(curl)-functionsand the standard bubble function techniques, we derive the a posteriori error estimates for the proposederror estimators. Then we exploit the convergence properties of the orthogonal πΏ2-projections and themesh-size functions to demonstrate that the sequences of the discrete states and controls generated bythe adaptive algorithm converge strongly to the exact solutions of the state and control in the energy-norm and πΏ2-norm, respectively, by first achieving the strong convergence towards the solution to alimiting control problem. Three-dimensional numerical experiments are also presented to confirm ourtheoretical results and the quasi-optimality of the adaptive edge element method.
Mathematics Subject Classification. 65K10, 65N12, 65N15, 65N30, 49J20.
Received August 8, 2020. Accepted August 17, 2021.
1. Introduction
Many electromagnetic simulation problems involve the following H(curl)-elliptic equation [13,44]:
curl(πβ1curly
)+ πy = f in Ξ©, (1.1)
where π and π are the electric permittivity and the magnetic permeability, respectively. This problem is alsoencountered when the implicit time-stepping scheme is used for solving the full time-dependent Maxwell system[31]. In this work, we are interested in a relevant optimal control problem, namely to find a specially designedexternal current source profile so that the resulting electromagnetic field achieves an optimal target design. Thisoptimal control problem has direct applications in many areas, such as the induction heating, the magneticlevitation and the electromagnetic material designs. We refer the readers to [12, 25, 63, 64] for some recentresults on the theory and applications of electromagnetic optimal control problems.
Keywords and phrases. Constrained optimal control, Maxwellβs equations, a posteriori error estimates; adaptive edge elementmethod, convergence analysis of adaptive algorithm.
1 Department of Mathematics, Duke University, Durham, NC 27708, USA.2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.*Corresponding author: [email protected]
c The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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2014 B. LI AND J. ZOU
The main purpose of this work is to design and analyse an adaptive edge element method for a class of optimalcontrol problems with the applied current density u being the control variable and the magnetic potential ybeing the state variable. Let yπ and uπ be our target magnetic field and control, and Uππ denote the unboundedclosed convex admissible set:
u β L2(Ξ©); u β₯ π a.e. in Ξ©, (1.2)
where π is some obstacle function with a certain regularity. Then we can write the control problem as aconstrained minimization problem: find a pair (y*,u*) β H0(curl, Ξ©)ΓUππ to minimize the quadratic objectivefunctional:
π½(y,u) :=12βcurl yβ yπβ20,Ξ© +
πΌ
2βuβ uπβ20,Ξ©, (1.3)
subject to the H(curl)-elliptic equation:
curl(πβ1curl y
)+ πy = f + u in Ξ©. (1.4)
By considering a simple transformation u = u β π, we may set π in (1.2) to zero in our subsequent analysisfor the sake of simplicity, that is, the admissible set shall take the form:
Uππ :=u β L2(Ξ©); u β₯ 0 a.e. in Ξ©
.
In this work, we are interested in the numerical study of the practically important situation where the localsingularities may be expected for the solution to the optimal control problem (1.3) and (1.4), due to the possibleirregular geometry of the domain Ξ©, the (possibly large) jumps across the interface between two different physicalmedia and the non-smooth source terms (cf. [19, 20]). In these cases, one typically needs to solve the problemon a very fine mesh to fully resolve the singularities, while it may be computationally expensive on account ofthe exponential growth of the number of degrees of freedom (DoFs). To balance the computational cost andthe numerical accuracy, it has proved to be more promising to refine the mesh adaptively when the numericalaccuracy is insufficient.
The theory of adaptive finite element methods (AFEMs), due to the pioneer work of Babuska and Rheinboldt[7] in 1978 for elliptic boundary value problems, has become very popular and well developed in the past fourdecades; see [2, 48] and the references therein for an overview. The relevant research in the case of Maxwellβsequations, which dated back to Beck et al. [10], has also reached a mature level nowadays (cf. [34, 53, 67, 68]).However, the theory of AFEMs for PDE-constrained optimization problems is still a work in progress. Recently,Gong and Yan [27] proved the convergence and quasi-optimality of AFEMs for an elliptic optimal controlproblem with the help of the variational discretization and the duality argument, and in this setting there is noneed to mark the data oscillations as in [26, 29]. In [42, 43], the authors considered the elliptic optimal controlproblems with integral control constraints and gave a rigorous proof of the convergence and the quasi-optimality.People may find more recent advances on AFEMs for the control problems in [5, 11,28,38].
The first contribution towards the adaptive methods for optimal control problems of Maxwellβs equationswent back to Hoppe and Yousept [35]. An AFEM was designed for the same model problem (1.3), (1.4), andthe a posteriori error estimation was also presented, including the reliability and efficiency, with Nedelecβs edgeelement approximations for both the control and the state. But the convergence of the adaptive method wasnot established in [35]. Later, Pauly and Yousept [49] considered an optimal control problem of first-ordermagnetostatic equations and presented the a posteriori error analysis. Meanwhile, Xu and Zou [61] consideredthe adaptive approximation for an optimal control problem associated with an electromagnetic saddle-pointmodel and proved the convergence of discrete solutions.
The current work is a continuation and further improvement of the work [35], where a piecewise linearH(curl)-conforming edge element was used to approximate the optimal control variable u. One crucial mo-tivation of our work is based on the observation that the optimal control u* is generally of low regularity,at most H1 (still under some reasonable conditions) (cf. Prop. 2.1). Therefore we can not expect a betterπΏ2-approximation accuracy of the control using any higher-order approximation than the piecewise constant
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2015
one (i.e., the simplest, and also cheapest, approximation). Because of this, we shall use the piecewise constantfunctions to approximate the control variable u, and propose a new adaptive algorithm with error indicatorsincorporating the residual-type error estimators and the lower-order data oscillations. As pointed out in [39,40],the precise structure of the admissible set and the way we discretize the distributed control problem are thecritical ingredients when designing an adaptive algorithm, which often influence the analysis of the resultingAFEM essentially. It turns out that in our case here, due to the pointwise unilateral constraints in the admissibleset Uππ, the inconsistency between the discrete spaces of the state and control as well as the low regularityof the given data, the data oscillation plays an important role in the performance of the algorithm and hasto be considered in the error estimates (as well as the marking strategy) to guarantee the convergence of thealgorithm. We can readily notice that such a change in the approximation of the control can significantly reducethe number of DoFs and make our algorithm much easier to implement than the algorithm in [35], while stillpreserving the same numerical accuracy; see Section 5 for the detailed discussions and the implementation issueon the algorithm. To theoretically justify the effectiveness of our adaptive algorithm, we use a Clement-typequasi-interpolation operator established by Schoberl [53] and the bubble functions [2] to derive the a posteriorierror estimates (reliability and efficiency estimates) of the new error estimators. It is worth mentioning thatthere is a technical issue in [35] when deriving the optimality system for the discrete optimization problem, thatis, the first-order optimality condition (2.8d) there cannot imply its complementarity condition (2.10) as theedge element discretization was adopted for the control (hence part of the estimates provided in [35] needs somenecessary modifications). This is another motivation of the current work. Instead, in our a posteriori analysis, weshall directly cope with the original forms of the first-order optimality conditions by using the contraction prop-erties of πΏ2-projections without the help of the complementarity problems (although they hold in our setting),so that our estimates can be established in a rigorous manner. Apart from the aforementioned contributionstowards the algorithmic aspects, another main contribution of this work is the strong convergence of the finiteelement solutions generated by the proposed adaptive method. This work appears to be the first one on theconvergence analysis of AFEMs applied to the electromagnetic optimal control problem involving a variationalinequality structure. In the finite element analysis provided in [63] for a quasilinear H(curl)-elliptic optimalcontrol problem, the discrete compactness of the Nedelec edge elements [36] is the main tool in establishingstrong convergence. However, the discrete compactness can only be applied to the discrete divergence-free se-quence, and it is still unknown whether it is true for adaptively generated meshes. During the course of ourconvergence analysis, we shall borrow some techniques from the nonlinear optimization to avoid the need ofdiscrete compactness and exploit some strategies with limiting spaces, which was initially adopted in [9] andthen developed systemically in [45]. To overcome the difficulties arising from the structure of the constraint setand its discretization, we establish a convergence property of πΏ2-projections associated with the meshes (cf.Prop. 4.2), which connects the convergence behaviors of the adaptive meshes and the discrete spaces. It furtherallows us to define a limiting optimization problem and characterize the limit point of the discrete controlsand states (cf. Thms. 4.4 and 4.5). Hereafter, we show that the maximal error estimators and the residualscorresponding to the sequence of adaptive states and adjoint states vanish (cf. Lems. 4.6 and 4.7), where we usea generalized convergence result of the mesh-size functions (cf. (4.4)), instead of introducing the buffer layer(cf. [61]) between the meshes at different levels, so that our proof can be significantly simplified. By meansof these auxiliary results and again the properties of πΏ2-projections, we are able to prove that the limit pointof the discrete triplets (y*π,p*π,u*π) solves the variational inequality associated with the control problem (cf.Thm. 4.8). Hence, the strong convergence of Algorithm 1, i.e., Theorem 4.1, follows immediately.
This work is organized as follows. In Section 2, we briefly review the electromagnetic optimal control problemand consider its finite element approximation. Then we propose the a posteriori error estimator and design anadaptive algorithm. We show that the error estimator is both reliable and efficient in Section 3. After that,we address the issue of the convergence of the adaptive solutions in Section 4. Finally, some numerical resultsare presented in Section 5 to illustrate the theoretical results and indicate the effectiveness and robustness ofthe adaptive approach versus the uniform mesh refinement. We end this section with a shorthand notation:π₯ . π¦ for π₯ β€ πΆπ¦ for some generic constant πΆ that is independent of the mesh size, but may depend on other
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2016 B. LI AND J. ZOU
quantities, e.g., π, πβ1, π and the shape regularity of the initial triangulation T0. If π₯ β€ πΆπ¦ and π₯ β₯ πΆπ¦ holdsimultaneously, we denote it simply by π₯ β π¦.
2. The optimal control problem and its discretization
We start by introducing the standard notation for Sobolev spaces and the involved physical parameters (cf.[1,44]). Let Ξ© β R3 be a bounded polyhedral domain with polyhedral Lipschitz subdomains Ξ©π, 1 β€ π β€ π, suchthat
Ξ©π β© Ξ©π = β for π = π and Ξ© =πβ
π=1
Ξ©π .
Let πΊ be an open bounded subset of Ξ© with the Lipschitz continuous boundary ππΊ and n being its unitoutward normal vector. For any real π > 0, we define the Hilbert space π»π (πΊ) (resp.Hπ (πΊ) := π»π (πΊ, R3)) forSobolev scalar functions (resp. vector fields) of order π with an inner product (Β·, Β·)π ,πΊ and a norm βΒ·βπ ,πΊ. Wealso introduce the spaces:
H(curl, πΊ) = v β L2(πΊ); curl v β L2(πΊ) and H(div, πΊ) = v β L2(πΊ); divv β πΏ2(πΊ),
equipped with the norm βvβcurl,πΊ = (βvβ20.πΊ + βcurl vβ20,πΊ)1/2 and βvβdiv,πΊ = (βvβ20.πΊ + βdivvβ20,πΊ)1/2, respec-tively. Here and in what follows, a bold typeface is used to indicate a vector-valued function. The tangentialtrace mapping πΎπ‘(u) := nΓ u and the normal trace mapping πΎπ(u) := u Β·n are well-defined on H(curl, πΊ) andH(div, πΊ), respectively. Then the zero tangential trace space can be introduced by
H0(curl, πΊ) = v β H(curl, πΊ); πΎπ‘(v) = 0 on ππΊ.
We will also use the fractional order curl-space
Hπ (curl, Ξ©) = u β Hπ (Ξ©); curlu β Hπ (Ξ©), π > 0, (2.1)
equipped with the norm βΒ·βπ»π (curl,Ξ©) := βΒ·βπ ,Ξ© +βcurlΒ·βπ ,Ξ©. In what follows, we write V for the most frequentlyused Sobolev space H0(curl, Ξ©).
We are now well-prepared for the mathematical formulation of the control problem. In this work, we focuson the following constrained minimization problem:
minuβUππ
π½(u) :=12βcurly(u)β yπβ20,Ξ© +
πΌ
2βuβ uπβ20,Ξ©, (2.2)
where the state y(u) β V is the solution to the variational system:
(πβ1curly, curlπ)0,Ξ© + (πy,π)0,Ξ© = (f + u,π)0,Ξ© β π β V. (2.3)
We assume that the given source term f β L2(Ξ©) satisfies
f |Ξ©π β H(div, Ξ©π) and πΎπ(f) β L2(πΞ©π), 1 β€ π β€ π, (2.4)
while the target magnetic field yπ and the target control uπ satisfy
yπ β H0(curl, Ξ©) and uπ β L2(Ξ©). (2.5)
We remark that although f is not in H(div) globally in Ξ©, namely f /β H(div, Ξ©), div f is well-defined on each Ξ©π
(hence almost everywhere on Ξ©) by the assumption (2.4). In what follows, we write div f forβπ
π=1 div fπΞ©π, by
abuse of notation, where πΞ©π is the characteristic function of Ξ©π. It is also clear that the weak divergence of f isglobally defined on Ξ© if and only if the jump of the normal trace [πΎπ(f)]Ξ across the interfaces Ξ := βͺπ
π=1πΞ©πβπΞ©
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2017
vanishes. The physical parameters π and π are supposed to be polynomials on each subdomain Ξ©π (hencepiecewise polynomials on Ξ©) and satisfy π(π₯) β₯ π0 > 0 and π(π₯) β₯ π0 > 0. The real πΌ > 0 is a stabilizationparameter. We define the symmetric bilinear form π΅(Β·, Β·) associated with the state equation (2.3):
π΅(u, v) :=(πβ1curlu, curlv
)0,Ξ©
+ (πu, v)0,Ξ©.
It readily follows from the assumptions of the physical parameters that π΅ is continuous and coercive, thusdefining an inner product on H(curl, Ξ©) whose induced energy norm βΒ·βπ΅ :=
βπ΅(Β·, Β·) is equivalent to βΒ·βcurl,Ξ©.
By the direct method in the calculus of variations (see, for instance, [58], Thms. 2.14β2.16) and noting thatthe cost functional π½(u) is weakly lower semicontinuous and strictly convex, we have that there always exists aunique minimizer u* β Uππ to the optimization problem. Then the first-order optimality condition yields(
curly(u*)β yπ, curly(u)β curly(u*))0,Ξ©
+ πΌ(u* β uπ,uβ u*
)0,Ξ©
β₯ 0 β u β Uππ, (2.6)
since π½(u) is Frechet differentiable. In what follows, we shall write y* for y(u*) for short. If we introduce theadjoint state p β V associated with y β V by
π΅(p,π) =(curlyβ yπ, curlπ
)0,Ξ©
β π β V,
equation (2.6) can be equivalently written as the following optimality system in terms of the state and theadjoint state, as well as the control variable:β§βͺβͺβ¨βͺβͺβ©
π΅(y*,π) = (f + u*,π)0,Ξ© β π β V,
π΅(p*,π) =(curly* β yπ, curlπ
)0,Ξ©
β π β V,(p* + πΌ
(u* β uπ
),uβ u*
)0,Ξ©
β₯ 0 β u β Uππ.
(2.7a)
(2.7b)
(2.7c)
If we further define the Lagrange multiplier (or the adjoint control) for the variational inequality (2.7c):
βπ* = βp* β πΌ(u* β uπ), (2.8)
then (2.7c) can be reformulated in a compact form:
βπ* β ππΌUππ(u*), (2.9)
where ππΌUππ is the subdifferential of the indicator function of the admissible set Uππ [33]. We remark that π* isactually the Frechet derivative of the functional π½(u). We then conclude from (2.7c), or (2.9), that u* is nothingelse than the πΏ2-projection of βp*
πΌ + uπ on Uππ, i.e.,
u* = PUππ
(βp*
πΌ+ uπ
). (2.10)
Here and throughout this work, we denote by PπΈ the πΏ2-projection on the convex subset πΈ of L2(Ξ©). It is worthmentioning that PUππ in (2.10) can also be understood in the pointwise sense due to the unilateral form of Uππ:
u* = max0,βp*
πΌ+ uπ
a.e. in Ξ©,
where the maximum is taken componentwise. We now end our discussion on the continuous optimal controlproblem with some regularity results.
Proposition 2.1. Assume that π is constant and Ξ© is convex or of class πΆ1,1. If uπ β H1(Ξ©) and π β H(div, Ξ©),then the triplet (y*,p*,u*) satisfying the optimality system (2.7a)β(2.7c) has the regularity: u*,y*,p* β H1(Ξ©).If we further assume that π is constant, we have y*,p* β H1(curl, Ξ©).
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2018 B. LI AND J. ZOU
Proof. It is clear that (2.7b) is equivalent to the following equation:
curlπβ1curlp* + πp* = curl(curly* β yπ
)in Ξ©, nΓ p* = 0 on πΞ©, (2.11)
in the distributional sense. By taking the divergence on both sides of (2.11), we have div(πp*) = 0. Recall thatthe space H0(curl, Ξ©) β©H(div, Ξ©) is continuously imbedded in H1(Ξ©), if Ξ© is convex or of class πΆ1,1 (cf. [6]).Since π is assumed to be constant, we readily see p* β H0(curl, Ξ©)β©H(div, Ξ©) βΛ H1(Ξ©). Noting that PUππ is acontinuous mapping from H1(Ξ©) to H1(Ξ©) (cf. [24], Sect. 5.10, [37]), we obtain from (2.10) and uπ β H1(Ξ©) thatπ’* β H1(Ξ©). Then using the fact that f+u* β H(div, Ξ©), a similar argument applying to y* yields y* β H1(Ξ©).
If we further assume that π is constant, it follows from the equation (2.7a) and de Rham diagram ([44], (3.60))that curly* belongs to the space H(curl, Ξ©)β©H0(div, Ξ©), which is also continuously imbedded in H1(Ξ©). Hence,there holds curly* β H1(Ξ©), which gives y* β H1(curl, Ξ©) (see (2.1)). Similarly, we can conclude by (2.7b)that p* β H1(curl, Ξ©), under the assumption (2.5) that yπ β H0(curl, Ξ©). The proof is complete.
Remark 2.2. The above result essentially relies on the regularity theory for Maxwellβs equations (cf. [3, 4, 60,62]). Here, we consider the case where the coefficients are isotropic and constant, so that the proof is direct.By using the recent result ([3], Thm. 1), one can similarly obtain the following generalization. Assume thatπ is constant, π β π 1,3+πΏ(Ξ©) for some πΏ > 0 and Ξ© is of class πΆ1,1. If uπ β H1(Ξ©) and π β H(div, Ξ©) holdas above, then the solution (y*,p*,u*) to the optimality system (2.7a)β(2.7c) still satisfies u* β H1(Ξ©) andy*,p* β H1(curl, Ξ©).
From the above discussion about the regularity, we can see that if there are no additional assumptions onthe physical coefficients, the given data and the domain Ξ©, the optimal control u* should generally be of lowregularity (less regular than H1). Hence we may expect from the standard (β-version) FEM theory that onuniform meshes, the simplest piecewise constant approximation can already achieve the optimal approximationaccuracy, and any higher-order approximations can not improve the numerical accuracy of the optimal control.In fact, this is still true even if the optimal control u* has the H1-regularity, since we aim only at the πΏ2-errorof the control variable. This is one of the main motivations of the current work. We would like to further remarkthat if the solution is piecewise analytic (singularity is allowed on a lower dimensional manifold), one may usegeneral versions of FEM (e.g. βπ-FEM) to get better convergence rate, even the exponential rate [8, 54, 56, 57](but the concrete design and implementation of such algorithms would be typically difficult and beyond thescope of this work).
The rest of this section is devoted to the finite element discretization of the control problem. We consider afamily of conforming and shape regular triangulations Tβ of Ξ©, which coincide with the partition Ξ© =
βππ=1 Ξ©π
in the sense that Ξ©π =β
πβΞ©ππ . We set β := maxπβTβ
βπ , where βπ denotes the diameter of π . We also referto Fβ :=
βππ β© Ξ©; π β Tβ, the set of all the interior faces, as the skeleton of the triangulation Tβ. The
diameter of a face πΉ is denoted by βπΉ . In our algorithm, we use the lowest-order H(curl)-conforming edgeelement space of Nedelecβs first family with zero tangential trace:
Vβ =vβ β H0(curl, Ξ©); vβ|π = aπ Γ x + bπ with aπ , bπ β R3, π β Tβ
,
to approximate both the state and the adjoint state. For approximating the control variable, we employ thesimplest space of element-wise constant functions with respect to the triangulation Tβ:
Uβ =uβ β L2(Ξ©); uβ|π β π0(π ), π β Tβ
.
Here and in what follows, ππ(π ) is the space of polynomials of degree β€ π over π . The discrete admissible setUππ
β is then given byUππ
β := Uβ β©Uππ = uβ β Uβ; uβ β₯ 0.With the help of these notions, we formulate the discrete optimal control problem:
minimize π½(yβ,uβ) =12βcurlyβ β yπβ20,Ξ© +
πΌ
2βuβ β uπβ20,Ξ© (2.12)
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2019
over (yβ,uβ) β Vβ ΓUππβ
subject to π΅(yβ,πβ) = (f + uβ,πβ)0,Ξ© β πβ β Vβ. (2.13)
As in the continuous case, the existence and uniqueness of the solution to the discrete system (2.12), (2.13) canbe guaranteed, and the corresponding optimality system reads as follows: find (y*β,p*β,u*β) β Vβ ΓVβ ΓUππ
β
by solving β§βͺβͺβ¨βͺβͺβ©π΅(y*β,πβ) = (f + u*β,πβ)0,Ξ© β πβ β Vβ,
π΅(p*β,πβ) =(curly*β β yπ, curlπβ
)0,Ξ©
β πβ β Vβ,(Pβp
*β + πΌ
(u*β β uπ
β
),uβ β u*β
)0,Ξ©
β₯ 0 β uβ β Uππβ ,
(2.14a)
(2.14b)
(2.14c)
where Pβ denotes the πΏ2-projection PUβ:
(Pβv)π :=1|π |
β«π
v(x)dx : L2(Ξ©) β Uβ, (2.15)
and the approximate target control uπβ in (2.14c) is given by Pβu
π. Similarly, we denote by yβ(uβ) the solutionto (2.13) and introduce the discrete reduced cost functional:
π½β(u) :=12βcurlyβ(uβ)β yπβ20,Ξ© +
πΌ
2βuβ β uπβ20,Ξ©,
then its Frechet derivative is given by (one can actually show π*β β Uππβ , i.e., π*β β₯ 0; see (2.18))
π*β = Pβp*β + πΌ(u*β β uπ
β)β Uβ. (2.16)
We also see from (2.14c) and the relation PUππβ
= PUππPβ that
u*β = PUππβ
(βp*β
πΌ+ uπ
)= PUππ
(βPβp
*β
πΌ+ uπ
β
), (2.17)
which directly implies a pointwise representation for the discrete optimal control:
u*β = max0,βPβp
*β
πΌ+ uπ
β
. (2.18)
3. Adaptive algorithm and the a posteriori error analysis
In this section, we give a complete discussion of the adaptive edge element method for solving the H(curl)-elliptic control problem and derive the a posteriori error estimates. An adaptive finite element method typicallytakes the successive loops:
SOLVE β ESTIMATE β MARK β REFINE,
where the a posteriori error estimation (module ESTIMATE) is the core step in the design of an adaptivealgorithm, through which we can extract the information on the error distribution. For our algorithm and anal-ysis, we shall use the residual-type a posteriori error estimators in terms of numerically computable quantities,which include the element residuals ππ , π β Tβ, and the face residuals ππΉ , πΉ β Fβ:
π2π :=
βπ=1,2
(π(π)π¦,π
)2
+β
π=1,2,3
(π(π)π,π
)2
, π2πΉ :=
βπ=1,2
(π(π)π¦,πΉ
)2
+(π(π)π,πΉ
)2
, (3.1)
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2020 B. LI AND J. ZOU
where π(π)π¦,π and π
(π)π.π are the element residuals defined by
π(1)π¦,π := βπ βf + u*β β curlπβ1curly*β β πy*ββ0,π , π
(2)π¦,π := βπ βdiv(fβ πy*β)β0,π ,
and
π(1)π,π := βπ βcurlyπ + curlπβ1curlp*β + πp*ββ0,π , π
(2)π,π := βπ βdiv(πp*β)β0,π , π
(3)π,π := βp*β β Pβp
*ββ0,π ,
while π(π)π¦,πΉ and π
(π)π,πΉ are the face residuals defined by
π(1)π¦,πΉ := β
1/2πΉ β[πΎπ‘(πβ1curly*β)]πΉ β0,πΉ , π
(2)π¦,πΉ := β
1/2πΉ β[πΎπ(f + u*β β πy*β)]πΉ β0,πΉ ,
π(1)π,πΉ := β
1/2πΉ β[πΎπ‘(βπβ1curlp*β + curly*β)]πΉ β0,πΉ , π
(2)π,πΉ := β
1/2πΉ β[πΎπ(πp*β)]πΉ β0,πΉ .
Here [Β·]πΉ := Β·|β β Β·|+ stands for the jump across the face πΉ , where the subscripts Β± denote the limits takenfrom outside and inside πΉ β ππ , respectively. We remark that it may not be easy to see how to define theseterms at the first glance, but we shall see that they are generated naturally in the reliability analysis. For easeof exposition, we denote by
π2β(π ) = π2
π +12
βπΉβππβ©Ξ©
π2πΉ ,
the residual-type error indicator associated with an element π . We should note that there are no face residualson the boundary of domain since the state equation satisfies the homogeneous boundary condition. For the aposteriori error estimates and the convergence analysis, a lower-order data oscillation related to uπ is needed:
osc2β
(uπ)
:=β
πβTβ
osc2π
(uπ), oscπ
(uπ)
:= βuπ β uπββ0,π , π β Tβ.
Some higher-order data oscillations associated with yπ and f shall also be involved:
osc2β
(yπ)
:=β
πβTβ
osc2π
(yπ)
with oscπ (yπ) := βπ βcurl(yπ β yπ
β
)β0,π , π β Tβ,
and
osc2β(f) :=
βπβTβ
osc2π (f)
with
oscπ (f) := βπ βfβ fββdiv,π +β
πΉβππβ©Ξ©
β1/2πΉ β[πΎπ(fβ fβ)]πΉ β0,πΉ , π β Tβ,
where we assume that yπβ β Vβ and fβ β Uβ are some approximations of yπ and f, respectively. In contrast to
the element residuals and face residuals associated with the discrete solutions, the data oscillation oscβ
(uπ)
istypically of lower order for a non-smooth target control uπ, and at most of π(β), by the Poincare inequality,even if uπ has a certain regularity. Meanwhile, the data oscillations oscβ
(yπ)
and oscβ(f) are of the same orderas the residuals, and shall be of higher order if the data yπ and f have additional regularities. Therefore, wecould replace yπ and f in the element residuals π
(π)π¦,π , π = 1, 2 and π
(1)π,π , as well as some face residuals, with yπ
β
and fβ for ease of implementation without any influence on the performance of the algorithm and any essentialchange of the analysis.
As we shall see, since the inconsistent discrete spaces are used and no additional regularity assumptions areadded on the data, the lower-order data oscillations may have a significant contribution to the total error so that
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2021
a single residual-type error estimator πβ is not enough to capture the error distribution accurately. Therefore, todesign an efficient adaptive algorithm for our problem, it is necessary to take into account the data oscillations[26, 29]. In view of this, we introduce a new mixed error indicator, by incorporating the lower-order dataoscillation oscπ
(uπ)
into the residual-type error estimator πβ,
π2β =
βπβTβ
π2β(π ) with π2
β(π ) = π2β(π ) + osc2
π
(uπ). (3.2)
We are now in a position to present the algorithm (see Algorithm 1 below), based on the error estimator πβ.In what follows, we shall use the iteration index π to indicate the dependence of a variable or a quantity on aparticular mesh generated in the adaptive process, for instance, we write ππ(π ) for πβ(π ) to emphasize that weare considering the error estimator πβ(π ) on the mesh Tπ.
Algorithm 1. Adaptive edge element method.1: Specify a shape regular initial mesh T0 on Ξ© and set the iteration index π := 0.2: (SOLVE) Compute the numerical solution (y*π,p*π u*π) to the discrete optimality system (2.14a)β(2.14c) on the mesh
Tπ.3: (ESTIMATE) Compute the error estimator ππ(π ) defined in (3.2) for each element π β Tπ.
4: (MARK) Mark a subset Mπ β Tπ containing at least one element ππ that satisfies
ππ
(ππ
)= max
πβTπ
ππ(π ). (3.3)
5: (REFINE) Refine elements in Mπ and other necessary elements by bisection to generate the smallest conformingmesh Tπ+1 with Tπ+1 β©Mπ = β .
6: Set π = π + 1 and go to Step 2 until the preset stopping criterion is met.
Several further remarks concerning each module in Algorithm 1 are in order. First, in the module SOLVE, weare required to solve a large-scale quadratic optimization problem with discretized PDE-constraints efficiently.It is nowadays a very active research area, and many high-performance algorithms and preconditioners havebeen developed for this purpose (cf. [16, 50β52,55]).
Second, to guarantee the convergence of the algorithm, we only need the condition (3.3) in the moduleMARK that the marked elements contain an element with the largest error indicator, which can be met bymany popular marking strategies, such as the maximum strategy [7], the equidistribution strategy [23] and theDorflerβs strategy [21]. In our numerical experiments (see Sect. 5), we shall use the Dorflerβs strategy to selectthe marked elements Mπ with minimal cardinality such that for a given π β (0, 1), there holdsβ
πβMπ
π2π(π ) β₯ π
βπβTπ
π2π(π ). (3.4)
Third, we only include the data oscillation oscβ
(uπ)
in the definition of πβ since it is a lower-order term thatmay provide a dominant error contribution among all the data oscillations. On the other hand, the behavior ofthe optimal control u* relies largely on the properties of the obstacle function π, whose information is furthertransferred to uπ. We hence expect that the approximation ability of Uβ associated with the current mesh Tβ
for uπ can directly influence the performance of our AFEM; see also [26,29] for related discussions. In the casewhere π is a constant function and uπ = 0, oscβ
(uπ)
vanishes and πβ becomes the standard residual-type errorestimator πβ.
Finally, for the module REFINE, all elements of Mπ are bisected at least once, and some additional elementsin TπβMπ may also need to be subdivided in order to generate a sequence of uniformly shape regular andconforming meshes Tππβ₯0; see [17] and Section 4 of [48] for the detailed mesh refinement algorithm and the
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2022 B. LI AND J. ZOU
necessary assumptions on the initial triangulation T0. Such a refinement process ensures that all the genericconstants involved in the inequalities below depend only on the shape regularity of the initial mesh and thegiven data.
We shall next present the a posteriori error analysis based on the error estimator πβ (3.2), including both thereliability and efficiency estimates, which is similar in spirit to the ones given in [35,53], but with several maindifficulties and differences as stated in the introduction, especially those caused by the inconsistency betweenthe discrete spaces of the state and control.
3.1. Reliability
In this section, we show that the error estimator πβ is reliable in the sense that it can provide an upper boundfor the total error between the true solution and the numerical solution:
βy*β β y*βcurl,Ξ© + βp*β β p*βcurl,Ξ© + βu*β β u*β0,Ξ©.
For this purpose, following [53], we introduce a helpful quasi-interpolation operator Ξ β. We start with thedefinition of the extended neighborhood Ξ©π for an element π β Tβ and fix some notations. Recall that theneighborhood Ξ©v of a vortex v is defined as the union of all the elements that contain the vertex v, and theextended neighborhood of a vertex v is given by Ξ©v :=
βvβ²βΞ©v
Ξ©vβ² . We define the extended neighborhood of anelement π by Ξ©π =
βvβπ
Ξ©v. An important consequence of the uniformly shape regularity of Tβ is that thecardinality of Ξ©π is uniformly bounded (cf. [48], Sect. 4.3):
maxπβTβ
#Ξ©π β€ πΆ(T0). (3.5)
The corresponding converse fact is that the collection of extended neighborhoods Ξ©π , π β Tβ, covers eachelement in Tβ finite times uniformly:
maxπβTβ
#
π β² β Tβ; π β Ξ©π β²
β€ πΆ(T0). (3.6)
Here the constants πΆ(T0) only depend on the initial mesh T0.
Lemma 3.1 ([53], Thm. 1). There exists an interpolation operator Ξ β : H0(curl, Ξ©) β Vβ such that for anyu β H0(curl, Ξ©), uβΞ βu has the decomposition:
uβΞ βu = βπ + z,
with π β π»10 (Ξ©), z β H1
0(Ξ©). Moreover, the following estimates hold
ββ1π βπβ0,π + ββπβ0,π . βuβ0,Ξ©π
, (3.7)
ββ1π βzβ0,π + ββzβ0,π . βcurluβ0,Ξ©π
. (3.8)
Since the optimal state y* and adjoint state p* satisfy the coupled system (2.7a)β(2.7c), the Galerkin orthog-onality, which is important for the a posteriori error analysis for the linear boundary value problems, does nothold here. To compensate it, we introduce the intermediate state and adjoint state, y(u*β) and p(u*β), by theequations:
π΅(y(u*β),π) = (f + u*β,π) β π β V, (3.9)
π΅(p(u*β),π) =(curly(u*β)β yπ, curlπ
)β π β V. (3.10)
If we use the variational discretization for the control variable, we can derive the following error equivalence(cf. [27, 30]):
βy(u*β)β y*ββcurl,Ξ© + βp(u*β)β p*ββcurl,Ξ© β βy*β β y*βcurl,Ξ© + βp*β β p*βcurl,Ξ© + βu*β β u*β0,Ξ©,
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2023
which allows us to directly conclude the reliability of the error estimator from the known result concerningMaxwellβs equations ([53], Cor. 2). If we use the edge element discretization for the control as in [35], the aboveerror equivalence still holds, up to the data oscillation oscβ
(uπ). However, this is not the case in our algorithm
since the discrete spaces of the control and state variables are different. Instead, we have the following result.
Lemma 3.2. Let the triplets (y*,p*,u*) and (y*β,p*β,u*β) be the solutions to (2.7a)β(2.7c) and (2.14a)β(2.14c),respectively. Then it holds that
βy*β β y*βcurl,Ξ© + βp*β β p*βcurl,Ξ© + βu*β β u*β0,Ξ©
. βy(u*β)β y*ββcurl,Ξ© + βp(u*β)β p*ββcurl,Ξ© + βPβp*β β p*ββ0,Ξ© + oscβ
(uπ).
Proof. By the well-posedness of the H(curl)-elliptic variational problem, we have
βy* β y(u*β)βcurl,Ξ© . βuβ u*ββ0,Ξ©, (3.11)βp* β p(u*β)βcurl,Ξ© . βcurly* β curly(u*β)β0,Ξ© . βuβ u*ββ0,Ξ©, (3.12)
which, combined with the triangle inequality, reduce the proof of the lemma to the estimate of βu*β β u*β0,Ξ©.In view of (2.10) and (2.17), we get
βu* β u*ββ20,Ξ© β€(u* β u*β,βp* β Pβp
*β
πΌ+ uπ β uπ
β
)0,Ξ©
, (3.13)
by the contraction property of πΏ2-projections ([15], Prop. 5.3). Moreover, we can deduce, by taking π =p* β p(u*β) in (3.9) and π = y* β y(u*β) in (3.10), that
(u* β u*β,p* β p(u*β))0,Ξ© = βcurly* β curly(u*β)β20,Ξ© β₯ 0. (3.14)
Combining (3.14) with (3.13) helps us obtain
βu*β β u*β0,Ξ© . βPβp*β β p*ββ0,Ξ© + βp*β β p(u*β)β0,Ξ© + βuπ
β β uπβ0,Ξ©,
which completes the proof of the lemma.
With the above preparations, we are now ready to prove the reliability of the error estimator.
Theorem 3.3. Let the triplets (y*,p*,u*) and (y*β,p*β,u*β) be the solutions to the continuous and discrete op-timality systems (2.7a)β(2.7c) and (2.14a)β(2.14c), respectively. Then we have the following reliability estimate:
βy*β β y*βcurl,Ξ© + βp*β β p*βcurl,Ξ© + βu*β β u*β0,Ξ© . πβ. (3.15)
Proof. By Lemma 3.2, it suffices to estimate βy(u*β) β y*ββcurl,Ξ© + βp(u*β) β p*ββcurl,Ξ© to obtain the reliabilityestimate (3.15). We first consider the estimate for the state variable y. Let ey be y(u*β) β y*β, and recall thenorm equivalence: βvβπ΅ β βvβcurl,Ξ©. We can derive, by the Galerkin orthogonality,
βeyβ2curl,Ξ© β π΅(ey, ey βΞ βey) = (f + u*β β πy*β, ey βΞ βey)0,Ξ© β(πβ1curly*β, curl(ey βΞ βey)
)0,Ξ©
. (3.16)
A direct application of Lemma 3.1 gives us the decomposition: eyβΞ βey = βπ+z with π β π»10 (Ξ©), z β H1
0(Ξ©).Substituting it into (3.16) and using integration by parts for each π , we have
(f + u*β β πy*β,βπ + z)0,Ξ© β(πβ1curly*β, curl(βπ + z)
)0,Ξ©
=β
πβTβ
β(div f, π)0,π +(f + u*β β πy*β β curlπβ1curly*β, z
)0,π
+ (div(πy*β), π)0,π
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2024 B. LI AND J. ZOU
+β
πΉβFβ
([πΎπ(f + u*β β πy*β)]πΉ , π)0,πΉ +([
πΎπ‘(πβ1curly*β)]πΉ, z)0,πΉ
. (3.17)
To proceed, by the scaled trace inequality ([48], Cor. 6.1):
βπ€β0,πΉ . ββ1/2πΉ βπ€β0,π + β
1/2πΉ ββπ€β0,π for πΉ β ππ, π€ β π»1(π ), (3.18)
and estimates (3.7) and (3.8), we obtain from (3.16) and (3.17) and the definitions of the error estimators,
βeyβ2curl,Ξ© .β
πβTβ
(π(1)π¦,π + π
(2)π¦,π
)βeyβcurl,Ξ©π
+β
πβTβ
βπΉβππβ©Ξ©
(π(1)π¦,πΉ + π
(2)π¦,πΉ
)βeyβcurl,Ξ©π
. (3.19)
By the property (3.6), the desired estimate follows from (3.19) and the Cauchyβs inequality:
βy(u*β)β y*ββcurl,Ξ© . πβ. (3.20)
The error ep := p(u*β)β p*β for the adjoint state can be analysed similarly. We note
βepβ2curl,Ξ© βπ΅(ep, ep βΞ βep) + π΅(ep, Ξ βep),
and write ep βΞ βep = βπ + z by Lemma 3.1. Then some elementary calculations give us that
π΅(ep, ep βΞ βep) = π΅(p(u*β)β p*β,βπ + z)
.(curly*β β yπ β πβ1curlp*β, curl z
)0,Ξ©
β (πp*β,βπ + z)0,Ξ© + βy*β β y(u*β)βcurl,Ξ©βepβcurl,Ξ©.
Moreover, by equations (2.14b) and (3.10), we have
|π΅(ep, Ξ βep)| . βy*β β y(u*β)βcurl,Ξ©βepβcurl,Ξ©.
Then we can derive by using integration by parts and the above estimates that
βepβ2curl,Ξ© .(curly*β β yπ β πβ1curlp*β, curl z
)0,Ξ©
β (πp*β,βπ + z)0,Ξ© + βy*β β y(u*β)βcurl,Ξ©βepβcurl,Ξ©
=β
πβTβ
(βcurlyπ β curlπβ1curlp*β β πp*β, z)0,π + (div(πp*β), π)0,π
ββ
πΉβFβ
([πΎπ‘(curly*β β πβ1curlp*β)]πΉ , z)0,πΉ β ([πΎπ(πp*β)]πΉ , π)0,πΉ
+ βy*β β y(u*β)βcurl,Ξ©βepβcurl,Ξ©,
which, by (3.20), the trace inequality (3.18) and Lemma 3.1, gives
βp(u*β)β p*ββcurl,Ξ© . πβ. (3.21)
Combining estimates (3.20) and (3.21) with Lemma 3.2 and the definition of πβ completes the proofof (3.15).
3.2. Efficiency
In this section, we consider the efficiency estimate, which is another aim of the a posteriori error analysis. Forthis, we need the so-called bubble functions, which plays a similar role to the cut-off functions and can help usestimate the local errors. As we shall see soon, when we deal with the divergence parts of the residual-type errorestimator πβ (i.e., π(2)
π¦,π , π(2)
π,π , π(2)
π¦,πΉ and π(2)
π,πΉ ), the higher-order bubble functions have to be used to ensure thevanishing boundary traces of some terms. Moreover, the curl structures in the right-hand sides of the adjoint
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2025
equations (2.7b) and (2.14b) also need our special and careful treatment. These important points were notaddressed in [35].
For the readerβs convenience, we next briefly review the definition of the bubble functions and some basicresults, see [2, 59] for a comprehensive introduction of this topic. We define the bubble function for an elementπ β Tβ by ππ (x) = 256Ξ 4
π=1πππ (x), x β π , where ππ
π , 1 β€ π β€ 4, are the barycentric coordinate functionsassociated with four vertices of π β Tβ. Similarly, the bubble function for a face πΉ β Fβ is given by ππΉ |π (x) =27Ξ 3
π=1ππΉπ (x), x β π β ππΉ . Here, ππΉ := π β Tβ; πΉ β ππ is the element pair for a face πΉ β Fβ, and ππΉ
π arethe barycentric coordinate functions associated with the vertices of the face πΉ β Fβ, which can be naturallyextended to ππΉ . To extend the face residuals defined on πΉ to ππΉ , we introduce the extension operator as follows.We first define the operator : πΆ(πΉ ) β πΆ(π ) on the reference element π in R3 by
[π](, π¦, π§) := π(, π¦),
where πΉ is the face of π lying on the (, π¦)-plane. By using the affine mapping πΉπ (x) = π΄π x+aπ : π β π β ππΉ ,the general extension operator πΈ : πΆ(πΉ ) β πΆ(ππΉ ) can be introduced by
πΈ[π]|π = [π β πΉπ ] β πΉβ1π , π β ππΉ , (3.22)
where the mappings πΉπ , π β ππΉ , are chosen such that πΉ is mapped to πΉ and πΈ[π] is well-defined on ππΉ andcontinuous. The next lemma summarizes the important properties of the bubble functions, which can be easilyverified by the equivalence of norms in a finite-dimensional linear space and the standard scaling argument.
Lemma 3.4. Let π be a positive integer and π be a positive real number. For any π β Tβ and πΉ β Fβ, thereholds
βπβ0,π . βππ π πβ0,π β€ βπβ0,π , βπβ0.πΉ . βππ
πΉ πβ0.πΉ β€ βπβ0,πΉ , (3.23)
for all π β ππ(π ) and π β ππ(πΉ ), and
β1/2πΉ βπβ0,πΉ . βππ
πΉ πΈ(π)β0,π . β1/2πΉ βπβ0,πΉ , (3.24)
for π β π€πΉ and π β ππ(πΉ ).
We are now in a position to state and prove the main result of this section.
Theorem 3.5. Let the triplets (y*,p*,u*) and (y*β,p*β,u*β) be the solutions to the continuous and discreteoptimality systems (2.7a)β(2.7c) and (2.14a)β(2.14c), respectively, and the multipliers π* and π*β be given by(2.8) and (2.16). Then we have the efficiency estimate:
πβ . βy*β β y*βcurl,Ξ© + βp*β β p*βcurl,Ξ© + βu*β β u*β0,Ξ© + βπ*β β π*β0,Ξ©
+ oscβ
(yπ)
+ oscβ(f) + oscβ
(uπ). (3.25)
Before we start our proof, we remark that it is necessary to consider the error of the multiplier βπ*ββπ*β0,Ξ©
here in order to estimate πβ, in comparison with [35], since the additional error estimator π(3)
π,π is included inπβ. We start with the following local efficiency estimate:
π(3)π,π = βp*β β p* + p* β Pβp
*ββ0,π
β€ βπ* β π*β β πΌ(u* β u*β β uπ + uπ
β
)β0,π + βp*β β p*β0,π
β€ βp* β p*ββ0,π + βπ* β π*ββ0,π + πΌ(βu* β u*ββ0,π + oscπ
(uπ))
,
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2026 B. LI AND J. ZOU
by the definitions of π*, π*β and π(3)π,π and the triangle inequality. Our proof proceeds by establishing more
local efficiency estimates for ππ , which are divided into the following four groups of estimates, for π β Tβ andπΉ β Fβ, β§β¨β© π
(1)π¦,π . βπ βu* β u*ββ0,π + βy* β y*ββcurl,π + oscπ (f),
π(1)π,π . βp
* β p*ββcurl,π + βy* β y*ββcurl,π + oscπ
(yπ),β§β¨β© π
(2)π¦,π . βu
* β u*ββ0,π + βy* β y*ββ0,π + oscπ (f),
π(2)π,π . βp
* β p*ββ0,π ,β§β¨β© π(1)π¦,πΉ . βπ βu* β u*ββ0,π€πΉ
+ βy* β y*ββcurl,π€πΉ+ π
(1)π¦,π+ + π
(1)π¦,πβ ,
π(1)π,πΉ . βp
* β p*ββcurl,π€πΉ+ βy* β y*ββcurl,π€πΉ
+ π(1)π,π+ + π
(1)π,πβ ,β§β¨β© π
(2)π¦,πΉ . βu
* β u*ββ0,π€πΉ+ βy* β y*ββ0,π€πΉ
+ π(2)π¦,π+ + π
(2)π¦,πβ + oscπ+(f),
π(2)π,πΉ . βp
* β p*ββ0,π€πΉ+ π
(2)π,π+ + π
(2)π,πβ ,
where π+ and πβ are two elements in ππΉ with πΉ = π+ β© πβ.
Proof. We give the proof of the above four groups of inequalities by the following four steps.
(1) We start with π(1)
π¦,π and readily see by the triangle inequality that
π(1)π¦,π β€ βπ βfβ + u*β β curlπβ1curly*β β πy*ββ0,π + oscπ (f). (3.26)
It is clear that ππ vanishes on ππ , and hence we can define zβ := ππ (fβ + u*β β curlπβ1curly*β β πy*β) βH0(curl, Ξ©). Using the estimate (3.23) with π = fβ + u*β β curlπβ1curly*β β πy*β and π = 1/2, we obtain
βfβ + u*β β curlπβ1curly*β β πy*ββ20,π β (fβ + u*β β curlπβ1curly*β β πy*β, zβ)0,π
= (fβ β f, zβ)0,π + (u*β β u*, zβ)0,π + π΅(y* β y*β, zβ). (3.27)
By estimates (3.26) and (3.27), and the inverse inequality:
βzββcurl,π . ββ1π βzββ0,π ,
as well as the norm equivalence: βzββ0,π β βfβ + u*β β curlπβ1curly*β β πy*ββ0,π , we can derive
π(1)π¦,π . βπ βu* β u*ββ0,π + βy* β y*ββcurl,π + oscπ (f).
The estimate of π(1)
π,π is similar. We note
π(1)π.π . βπ βcurlyπ
β + curlπβ1curlp*β + πp*ββ0,π + oscπ
(yπ), (3.28)
and define zβ := ππ (curlyπβ + curlπβ1curlp*β + πp*β) β H0(curl, Ξ©). Then a similar estimate as above
gives
βcurlyπβ + curlπβ1curlp*β + πp*ββ20,π . (curlyπ
β + curlπβ1curlp*β + πp*β, zβ)0,π
.(curlyπ + curlπβ1curlp*β + πp*β, zβ
)0,π
β (curly*, curl zβ)0,π
+(curlyπ
β β curlyπ, zβ
)0,π
+ (curl(y* β y*β), curlπ§β)0,π
(3.29)
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2027
. βcurlyπβ β curlyπβ0,π βzββ0,π + βy*β β y*βcurl,π βzββcurl,π
+ βp*β β p*βcurl,π βzββcurl,π , (3.30)
where in (3.29) we have used (curly*β, curl zβ)0,π = 0 from the fact that y*β is a first-order polynomial onπ , and in (3.30) we have used(
curlyπ + curlπβ1curlp*β + πp*β, zβ
)0,π
β (curly*, curl zβ)0,π
= π΅(p*β β p*, zβ) . βp*β β p*βcurl,π βzββcurl,π .
Further applying the inverse estimate for βzββcurl,π in (3.30) and recalling (3.28), we come to
π(1)π,π . βp
* β p*ββcurl,π + βy* β y*ββcurl,π + oscπ
(yπ).
(2) Define zβ := div(fβ β πy*β)π2π with βzββ0,π β βdiv(fβ β πy*β)β0,π . It is clear that βzβ is a polynomial on π
and vanishes on the boundary ππ , which gives βzβ β H0(curl, Ξ©). By a direct calculation, we have
(div(fβ β πy*β), zβ)0,π = (div(fβ β πy*β + u*β), zβ)0,π
= (div(fβ πy*β + u*β), zβ)0,π + (div(fβ β f), zβ)0,π
= (βπy* + πy*β + u* β u*β,βzβ)0,π + (div(fβ β f), zβ)0,π , (3.31)
where we have used the following observation in the last equality:
π΅(y*,βzβ) = (πy*,βzβ)0,Ξ© = (f + u*,βzβ)0,Ξ©,
which is from (2.7a) with the test function π = βzβ. Again, by Lemma 3.4 and the inverse estimate, wecan derive from the definition of π(2)
π¦,π and (3.31) that
π(2)π¦,π . βu
*β β u*β0,π + βy*β β y*β0,π + oscπ (f).
Likewise, for π(2)
π,π , taking zβ = div(πp*β)π2π and observing from (2.7b):
(πp*,βzβ)0,Ξ© =(curly* β yπ, curlβzβ
)0,Ξ©
= 0,
we can derive, by almost the same arguments as above, that
π(2)π,π . βp
* β p*ββ0,π .
(3) Since the lowest-order edge element is used for the discretization, curly*β is a piecewise constant vector.Then πΎπ‘(πβ1curly*β) is a polynomial defined on πΉ and can be extended to π€πΉ by the extension operatorπΈ introduced in (3.22). Define
zβ := ππΉ πΈ([
πΎπ‘
(πβ1curly*β
)]πΉ
)β H0(curl, Ξ©).
By the estimate (3.23) and integration by parts over ππΉ , we have(π(1)π¦,πΉ
)2
= βπΉ β[πΎπ‘
(πβ1curly*β
)]πΉβ20,πΉ β βπΉ
([πΎπ‘
(πβ1curly*β
)]πΉ, zβ
)0,πΉ
= βπΉ
(curlπβ1curly*β + πy*β β fβ u*β, zβ
)0.π€πΉ
β βπΉ
(πβ1curly*β β πβ1curly*, curl zβ
)0,π€πΉ
+ βπΉ (u*β β u*, zβ)0,π€πΉ+ βπΉ (πy* β πy*β, zβ)0,π€πΉ
. (3.32)
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2028 B. LI AND J. ZOU
The estimate (3.24) and the inverse estimate give us
β1/2πΉ β
[πΎπ‘
(πβ1curly*β
)]πΉβ0,πΉ β βzββ0,π€πΉ
, βcurl zββ0,π€πΉ. ββ1
πΉ βzββ0,π€πΉ. (3.33)
Combining (3.33) with the formula (3.32), we get
π(1)π¦,πΉ . π
(1)π¦,π+ + π
(1)π¦,πβ + βy*β β y*βcurl,π€πΉ
+ βπΉ βu*β β u*β0,π€πΉ.
For π(1)
π,πΉ , let zβ := ππΉ πΈ([
πΎπ‘
(βπβ1curlp*β + curly*β
)]πΉ
). By similar calculations, it follows that(
π(1)π,πΉ
)2
β βπΉ
([πΎπ‘
(βπβ1curlp*β + curly*β
)]πΉ, zβ
)0,πΉ
= βπΉ
(βcurlπβ1curlp*β β curlyπ β πp*β, zβ
)0,π€πΉ
β βπΉ
(βπβ1curlp*β + curly*β β yπ, curl zβ
)0,π€πΉ
+ βπΉ (πp*β, zβ)0,π€πΉ(3.34)
= βπΉ
(βcurlπβ1curlp*β β curlyπ β πp*β, zβ
)0,π€πΉ
+ βπΉ π΅(p*β β p*, zβ)
+ βπΉ (curl(y* β y*β), curl zβ)0,ππΉ, (3.35)
where we have used (curl curly*β, zβ)0,π = 0 for π β ππΉ in (3.34). Then, by the inverse estimate andLemma 3.4, a direct estimate leads to
π(1)π,πΉ . π
(1)π,π+ + π
(1)π,πβ + βp* β p*ββ0,π€πΉ
+ βy*β β y*βcurl,π€πΉ.
(4) For π(2)π¦,πΉ , we define zβ := π2
πΉ πΈ([πΎπ(fβ + u*β β πy*β)]πΉ ). It is easy to see that βzβ β H0(curl, Ξ©). Takingπ = βzβ in (2.7a) gives
(f + u* β πy*,βzβ)0,π€πΉ= 0. (3.36)
We note by the triangle inequality that
π(2)π¦,πΉ β€ β
1/2πΉ β[πΎπ(fβ + u*β β πy*β)]πΉ β0,πΉ + β
1/2πΉ β[πΎπ(fβ fβ)]πΉ β0,πΉ . (3.37)
Then we have, again by the property of the bubble functions (3.23) and integration by parts over ππΉ ,
β[πΎπ(fβ + u*β β πy*β)]πΉ β20,πΉ β ([πΎπ(fβ + u*β β πy*β)]πΉ , zβ)
0,πΉ
. (div(fβ πy*β), zβ)0,π€πΉ+ ([πΎπ(fβ β f)]πΉ , zβ)
0,πΉ
+ (f + u*β β πy*β,βzβ)0,ππΉ, (3.38)
where we can use (3.36) to rewrite the last term as
(f + u*β β πy*β,βzβ)0,ππΉ= (u*β β u*,βzβ)0,π€πΉ
+ (πy* β πy*β,βzβ)0,π€πΉ. (3.39)
Therefore, we obtain from (3.37) to (3.39), the Cauchyβs inequality, applying the inverse estimate toββzββ0,ππΉ
and the trace inequality to βzββ0,πΉ (cf. (3.18)) that
π(2)π¦,πΉ . βu
* β u*ββ0,π€πΉ+ βy* β y*ββ0,π€πΉ
+ π(2)π¦,π+ + π
(2)π¦,πβ + oscπ+(f),
where we have used the trivial bound that β1/2πΉ β[πΎπ(fβ fβ)]πΉ β0,πΉ β€ oscπ+(f). The estimate for π(2)
π,πΉ followsfrom the same (even simpler) argument. In fact, we can define zβ = π2
πΉ πΈ([πΎπ(πp*β)]πΉ ), which impliesβzβ β H0(curl, Ξ©) and, by (2.7b), (πp*,βzβ)0,Ξ© = 0. Then, a typical calculation gives
β[πΎπ(πp*β)]πΉ β20,πΉ β ([πΎπ(πp*β)]πΉ , zβ)
0,πΉ= (div(πp*β), zβ)0,π€πΉ
+ (πp*β β πp*,βzβ)0,π€πΉ,
which yields, by Lemma 3.4 and the inverse estimate,
π(2)π,πΉ . βp
* β p*ββ0,π€πΉ+ π
(2)π,π+ + π
(2)π,πβ .
Theorem 3.5 follows now by adding up the above local efficiency estimates over all π β Tβ.
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2029
4. Convergence
We devote this whole section to establish our main result that the sequence of adaptively generated finiteelement solutions (y*π,p*π,u*π)πβ₯0 converges strongly to the true solution (y*,p*,u*).
Theorem 4.1. Let (y*π,p*π,u*π)πβ₯0 be the sequence of discrete triplets generated by the adaptive Algorithm 1and (y*,p*,u*) be the solution to the system (2.7a)β(2.7c). Then we have the strong convergences:
limπββ
βy*π β y*βcurl,Ξ© = 0, limπββ
βp*π β p*βcurl,Ξ© = 0 and limπββ
βu*π β u*β0,Ξ© = 0. (4.1)
As already pointed out in the introduction, the first step of the proof of convergence is the introduction ofa limiting minimization problem that characterizes the limit of the discrete solutions to the system (2.14a)β(2.14c) while the second step is to show that the solution to the limiting problem actually coincides with theone to (2.7a)β(2.7c). To do so, let us first state some helpful auxiliary results on the convergence behavior of theadaptive meshes Tππβ₯0. We associate each triangulation Tπ with a mesh-size function βπ β πΏβ(Ξ©), definedby βπ|π = βπ for π β Tπ. Thanks to the monotonicity of βππβ₯0, we are allowed to define a limiting mesh-sizefunction by the pointwise limit: βπ(x) β ββ(x), as π ββ. It is also known that the pointwise convergence ofβπ can be further improved to the uniform convergence ([48], Lem. 4.2):
limπββ
ββπ β ββββ,Ξ© = 0. (4.2)
We should note that ββ(x) β‘ 0 in general. If ββ(x) > 0 at some point x, there is an element π containing xand an index π(x) depending on x such that π β Tπ for all π β₯ π(x). This observation motivates us to split Tπ
into two classes of elements:T +
π :=βπβ₯π
Tπ and T 0π := TπβTπ
+. (4.3)
T +π consists of the elements in Tπ that are not refined after the πth iteration, while T 0
π β Tπ contains theelements that are refined at least once in the subsequent iterations. We are thus allowed to decompose thedomain Ξ© into two parts: Ξ©+
π :=β
πβT +π
π and Ξ©0π :=
βπβT 0
ππ . A direct application of (4.2) and the uniform
shape regularity of Tπ yields the following result (cf. [48], Cor. 7.1):
limπββ
ββπββ,Ξ©0π
= 0, (4.4)
where Ξ©0π :=
βπβT 0
π
Ξ©π is the extended neighborhood of Ξ©0π. We remark that this uniform convergence result
for the mesh-size functions is crucial for our subsequent analysis. We next give an interesting characterizationof the limiting behavior of πΏ2-projections Pππβ₯0 := PUπ
πβ₯0 with the help of the convergence propertyof the adaptive meshes Tππβ₯0, which establishes a connection between the limit of mesh-size functions, theπΏ2-projections and the limiting problem that we shall propose and deal with in the next section.
Proposition 4.2. Let Pππβ₯0 be the orthogonal πΏ2-projections (defined by (2.15)) associated with the adaptivemeshes Tππβ₯0 generated by Algorithm 1. Then for each f β L2(Ξ©), the limit of the sequence Pπfπβ₯0,denoted by Pβf, exists as π β β. Furthermore, the corresponding limiting operator Pβ is also an orthogonalπΏ2-projection with the range and kernel given by
ran(Pβ) = Uβ :=βπβ₯0
Uπ
πΏ2
and ker(Pβ) = Uβ₯β. (4.5)
Proof. It suffices to show that the limit of the sequence Pπf exists for all f β Cβπ (Ξ©) (infinitely differentiable
vector-valued functions with compact support in Ξ©), by the facts that the operators Pπ are uniformly boundedand the space Cβ
π (Ξ©) is dense in L2(Ξ©). Given two iteration indices π1, π2 with π2 > π1, define T 0π1,π2
:=
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2030 B. LI AND J. ZOU
Tπ1β(Tπ1 β©Tπ2), which is a subset of T 0π1 consisting of the elements that are refined between the π1th iteration
and π2th iteration. We then have, by the equivalent definition of Pπ in (2.15),
βPπ1fβ Pπ2fβ20,Ξ© = β
πβT 0π1,π2
βππβπ,ππβTπ2
(1|π |
β«π
f(x)dxβ 1|ππ|
β«ππ
f(x)dx)
πππ
2
0,Ξ©
=β
πβT 0π1π2
βππβπ,ππβTπ2
(1|π |
β«π
f(x)dxβ 1|ππ|
β«ππ
f(x)dx)2
|ππ|,
where πππ is the characteristic function of ππ. Recalling the limiting behavior of mesh-size functions βπ in(4.4), we have that for any πΏ > 0, there exists an index π(πΏ) depending on πΏ such that for all π > π(πΏ), thereholds ββπββ,Ξ©0
πβ€ πΏ. Combining it with the uniform continuity of f, we have that for any π > 0, there exists an
index π(π, πΏ) depending on π and πΏ such that for any integers π1, π2 satisfying π2 > π1 > π(π, πΏ), and for anyelements π β T 0
π1,π2, ππ β Tπ2 with ππ β π , it holds that
1|π |
β«π
f(x)dxβ 1|ππ|
β«ππ
f(x)dxβ€ π.
We hence have that Pπf is a Cauchy sequence in L2(Ξ©). Then it follows from the completeness of L2(Ξ©) thatthe limit of Pπf exists. We have proved that for any f β L2(Ξ©), Pβπ is well-defined, which further allows usto define the bounded linear operator Pβ on L2(Ξ©), by the uniform boundedness principle. We now show thatPβ is an orthogonal πΏ2-projection with the property (4.5). To do so, we first observe that for any g β L2(Ξ©),it holds that
(Pβf, g)0,Ξ© = limπββ
(Pπf, g)0,Ξ© = limπββ
(Pπg, f)0,Ξ© = (f, Pβg)0,Ξ©,
which implies that Pβ is self-adjoint. On the other hand, we have
β(P2β β Pβ
)fβ0,Ξ© = β
(P2β β PπPβ + PπPβ β P2
π + Pπ β Pβ)fβ0,Ξ©
β€ β(Pβ β Pπ)(Pβf)β0,Ξ© + βPπββ(Pβ β Pπ)fβ0,Ξ©
+ β(Pπ β Pβ)fβ0,Ξ© β 0 as π β 0,
that is, P2β = Pβ. Thus, we can conclude that Pβ is an orthogonal πΏ2-projection. To characterize its range
and kernel, we denote by P the orthogonal projection associated with the closed space Uβ defined in (4.5).By definition, there holds ran(P) = Uβ and ker(P) = Uβ₯
β. We readily see from the definition of Pβ thatran(Pβ) β Uβ, since every element in ran(Pβ) can be approximated by a sequence from
βπβ₯0 Uπ. Conversely,
to prove Uβ β ran(Pβ), we note that for any f β Uπ, Pπf = f holds for each π β₯ π. Then, letting π ββ givesus
Pβf = limπββ
Pπf = f,
which indicates Uπ β ran(Pβ). Since π is arbitrary, we readily seeβ
πβ₯0 UππΏ2
β ran(Pβ). The proof iscomplete.
4.1. The limiting problem
In order to find the limit point of the discrete triplets (y*π,p*π,u*π)πβ₯0, we first define several limiting spacesas the closure of the union of discrete spaces at each level:
Vβ :=βπβ₯0
Vπ
H(curl)
, Uππβ :=
βπβ₯0
Uπππ
πΏ2
. (4.6)
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2031
which immediately implies Uππβ = Uβ β©Uππ, where the space Uβ is defined in (4.5). Since Uππ
β is a convexsubset of L2(Ξ©), it is closed in the weak topology if and only if it is closed in the strong topology induced bythe norm. We hence have the following key lemma.
Lemma 4.3. Let uπ β Uπππ , π β₯ 0, be a sequence weakly converging to a u in L2(Ξ©). Then u β Uππ
β holds.
We now consider the following limiting problem defined on the limiting spaces:
minimize π½(yβ,uβ) =12βcurlyβ β yπβ20,Ξ© +
πΌ
2βuβ β uπβ20,Ξ© (4.7)
over (yβ,uβ) β Vβ ΓUππβ
subject to π΅(yβ,πβ) = (f + uβ,πβ)0,Ξ© β πβ β Vβ. (4.8)
The existence and uniqueness of the minimizer to the above problem can be obtained by the standard argumentsas the continuous and discrete cases, and the reduced cost functional for the limiting problem is defined by
π½β(uβ) :=12βcurlyβ(uβ)β yπβ20,Ξ© +
πΌ
2βuβ β uπβ20,Ξ©,
where yβ(uβ) denotes the solution to the equation (4.8) with the source uβ. The rest of this subsection isdevoted to proving that the limit point of (y*π,p*π,u*π)πβ₯0 exists and happens to be the solution to the limitingoptimization problem. As pointed out in the introduction, to compensate the lack of the discrete compactness,we shall investigate the weak limit of the sequence u*ππβ₯0 first, and then improve it to the strong convergence;see the next two theorems.
Theorem 4.4. Suppose that (u*π,y*π)πβ₯0 is the sequence of discrete optimal controls and states generated byAlgorithm 1, and (u*β,y*β) is the optimal control and state to the limiting optimization problem (4.7) and (4.8).Then, we have the following weak convergences: as π ββ,
u*ππ€ u*β in L2(Ξ©) and y*π
π€ y*β in H0(curl, Ξ©).
Proof. Our proof starts with a simple but important observation that the sequence u*π is bounded, which isnot a trivial fact due to the unboundedness of Uππ. Indeed, for a fixed u β Uππ, we have
πΌ
2βu*π β uπβ20,Ξ© β€ π½π(u*π) β€ π½π(Pπu) =
12βcurlyπ(Pπu)β yπβ20,Ξ© +
πΌ
2βPπuβ uπβ20,Ξ©
. βuβ20,Ξ© + βyπβ20,Ξ© + βuπβ20,Ξ©,
which gives the boundedness of u*π in L2(Ξ©). Then the boundedness of y*π follows immediately. Hence, bythe Banach-Alaoglu theorem, we can extract a subsequence (y*ππ
,u*ππ)πβ₯1 of (y*π,u*π)πβ₯0 such that u*ππ
π€ w
in L2(Ξ©) and y*ππ
π€ y in H(curl, Ξ©), as π tends to infinity, which further implies that y*ππ
and curly*ππ
weakly converge to y and curly, respectively, in L2(Ξ©) (since H(curl, Ξ©) is continuously imbedded in L2(Ξ©)and curl is a bounded linear operator from H(curl, Ξ©) to L2(Ξ©)). Moreover, Lemma 4.3 yields w β Uππ
β . NotingVπ β Vππ
for π β€ ππ and that there holds
π΅(y*ππ, vπ) = (f + u*ππ
, vπ)0,Ξ© for π β€ ππ and vπ β Vπ, (4.9)
we let π tend to infinity in (4.9) and obtain, by the weak convergences of u*ππ and y*ππ
,
π΅(y, vπ) = (f + w, vπ)0,Ξ© for all π β₯ 0 and vπ β Vπ.
Sinceβ
πβ₯0 Vπ is dense in Vβ, we readily have
π΅(y, vβ) = (f + w, vβ)0,Ξ© β vβ β Vβ (4.10)
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2032 B. LI AND J. ZOU
which means that (y,w) satisfies the constraint (4.8) of the limiting problem, that is, y = yβ(w).We claim that w is the minimizer u*β of the cost functional π½β over the set Uππ
β , namely,
π½β(w) β€ π½β(uβ) β uβ β Uππβ , (4.11)
which, by (4.10), also implies that the weak limit y of the sequence y*ππ is y*β. To prove the claim, we first
note
π½β(w) =12βcurlyβ(w)β yπβ20,Ξ© +
πΌ
2βwβ uπβ20,Ξ©
β€ lim infπββ
12βcurly*ππ
β yπβ20,Ξ© + lim infπββ
πΌ
2βu*ππ
β uπβ20,Ξ© = lim infπββ
π½ππ(u*ππ
), (4.12)
by the uniform boundedness principle and the weak converges of curly*ππ and u*ππ
to curlyβ(w) and w inL2(Ξ©). Clearly, by (4.12) and the fact that u*ππ
is the minimizer of π½ππ over Uππππ
, there holds, for any sequence,uπ β Uππ
π , π β₯ 0,π½β(w) β€ lim inf
πββπ½ππ
(u*ππ) β€ lim sup
πββπ½ππ
(u*ππ) β€ lim sup
πββπ½π(uπ). (4.13)
We next prove an auxiliary fact that for any uβ β Uππβ and a sequence uπ β Uππ
π for π β₯ 0, if βuπβuββ0,Ξ© β 0as π ββ, then
limπββ
βyβ(uβ)β yπ(uπ)βcurl,Ξ© = 0, (4.14)
which readily giveslim
πββπ½π(uπ) = π½β(uβ). (4.15)
This fact (4.15), along with (4.13), completes our proof of the claim (4.11). To prove (4.14), we note, by theassumption,
lim supπββ
βyβ(uβ)β yπ(uπ)βcurl,Ξ© β€ lim supπββ
(βyβ(uβ)β yπ(uβ)βcurl,Ξ© + βyπ(uβ)β yπ(uπ)βcurl,Ξ©)
. lim supπββ
( infvπβππ
βyβ(uβ)β vπβcurl,Ξ© + βuβ β uπβ0,Ξ©) = 0,
where we have used the Galerkin orthogonality and the density ofβ
πβ₯0 Vπ in Vβ.By the above arguments, we can conclude that for any subsequence (u*ππ
,y*ππ)πβ₯1 of (u*π,y*π)πβ₯0, we can
extract a subsequence weakly converging to (u*β,y*β) in L2(Ξ©) ΓH0(curl, Ξ©), which immediately yields theweak convergence of the whole sequence (u*π,y*π)πβ₯0:
u*ππ€ u*β in L2(Ξ©) and y*π
π€ y*β in H0(curl, Ξ©), as π ββ.
The proof is complete.
Thanks to the above results, we are ready to show the main theorem of this subsection.
Theorem 4.5. Under the same assumptions as in Theorem 4.4, there holds
limπββ
βu*π β u*ββ0,Ξ© = 0 and limπββ
βy*π β y*ββcurl,Ξ© = 0. (4.16)
Proof. We note from the claim (4.14) in the proof of Theorem 4.4 that the second convergence in (4.16) isa consequence of the first one. Hence, it suffices to prove the convergence of u*ππβ₯0. To this end, a directcalculation gives
βcurly*π β curly*ββ20,Ξ© + πΌβu*π β u*ββ20,Ξ©
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2033
= βcurly*π β yπ + yπ β curly*ββ20,Ξ© + πΌβu*π β uπ + uπ β u*ββ20,Ξ©
= βcurly*π β yπβ20,Ξ© + βcurly*β β yπβ20,Ξ© β 2(curly*π β yπ, curly*β β yπ
)0,Ξ©
+ πΌβu*π β uπβ20,Ξ© + πΌβu*β β uπβ20,Ξ© β 2πΌ(u*π β uπ,u*β β uπ
)0,Ξ©
,
which, by taking the upper limit on both sides and using Theorem 4.4, implies
2π½β(u*β) + lim supπββ
(βcurly*π β curly*ββ20,Ξ© + πΌβu*π β u*ββ20,Ξ©
)β€ lim sup
πββ2π½π(u*π). (4.17)
Then it follows thatπ½β(u*β) β€ lim sup
πββπ½π(u*π). (4.18)
We choose a sequence uπ β Uπππ , π β₯ 0, such that βuπ β u*ββ0,Ξ© β 0, as π β β, and then we can derive, by
(4.18) and (4.15), thatπ½β(u*β) β€ lim sup
πββπ½π(u*π) β€ lim
πββπ½π(uπ) = π½β(u*β). (4.19)
Combining the above estimate (4.19) with (4.17), we obtain the strong convergence of u*ππβ₯0.
It is easy to write the optimality system for the limiting problem by a standard argument:β§βͺβͺβ¨βͺβͺβ©π΅(y*β,πβ) = (f + u*β,πβ)0,Ξ© β πβ β Vβ,
π΅(p*β,πβ) =(curly*β β yπ, curlπβ
)0,Ξ©
β πβ β Vβ,(p*β + πΌ
(u*β β uπ
),uβ β u*β
)0,Ξ©
β₯ 0 β uβ β Uππβ ,
(4.20a)
(4.20b)
(4.20c)
and see that the sequence of discrete triplets (y*π,p*π,u*π)πβ₯0 converges strongly to the solution (y*β,p*β,u*β)to the limiting optimality system (4.20a)β(4.20c). We remark that the variational inequality (4.20c) will be usedin the next subsection.
4.2. Proof of convergence
We have proved the strong convergence of the discrete solutions (y*π,p*π,u*π) in Section 4.1, where we haveonly used the structure of the control problem, while the adaptive process does not have an essential involvement.We shall see that in the subsequent analysis, the error estimator πβ defined in (3.2) and the marking requirement(3.3) in Algorithm 1 play a crucial role. To complete the proof of Theorem 4.1, we start with the following keylemma.
Lemma 4.6. Let ππ β Tπ be one of the elements that achieve the maximum value of ππ(π ) over π β Tπ, i.e.,ππ
(ππ
):= max
πβTπ
ππ(π ). Then we have
limπββ
ππ
(ππ
)= 0. (4.21)
Proof. We start with the estimates for π(1)
π¦,π and π(1)
π¦,πΉ on a given mesh Tπ. A direct application of the inverseestimate and the triangle inequality gives
π(1)π¦,π . βπ βfβ0,π + βπ βu*ββ0,π + βcurly*ββ0,π + βπ βy*ββ0,π ,
and, by the assumption of π and the trace inequality (3.18), we have
π(1)π¦,πΉ . β
1/2πΉ β(curly*β)|ββ0,πΉ + β
1/2πΉ β(curly*β)|+β0,πΉ . βcurly*ββ0,ππΉ
.
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2034 B. LI AND J. ZOU
We now consider π(2)
π¦,π and π(2)
π¦,πΉ . Similarly, we have
π(2)π¦,π . βπ βdiv fβ0,π + βy*ββ0,π ,
and
π(2)π¦,πΉ . β
1/2πΉ β[πΎπ(f)]πΉ β0,πΉ + β
1/2πΉ β[u*β]πΉ β0,πΉ + β
1/2πΉ β[y*β]πΉ β0,πΉ (4.22)
by the triangle inequality. Then the trace inequality (3.18) gives
β1/2πΉ β[u*β]πΉ β0,πΉ + β
1/2πΉ β[y*β]πΉ β0,πΉ . βu*ββ0,ππΉ
+ βy*ββcurl,ππΉ, (4.23)
since for π β ππΉ , u*β|π is a constant vector and y*β|π β H1(π ) satisfiesβ
2|βy*β| = |curly*β| which can bedirectly checked by using y*β|π = aπ Γx+bπ for some aπ , bπ β R3. It is clear from (2.4) that β[πΎπ(f)]πΉ β0,πΉ > 0,only if πΉ β Ξ = βͺπ
π=1πΞ©πβπΞ©. Hence, it follows from (4.22) and (4.23) that
π(2)π¦,πΉ . β
1/2πΉ β[πΎπ(f)]πΉ β0,πΉβ©Ξ + βu*ββ0,ππΉ
+ βy*ββcurl,ππΉ.
The same analysis applies to the other terms in ππ. Then a careful but straightforward computation shows, forπ β Tπ,
ππ(π ) . βy*πβcurl,ππ+ βp*πβcurl,ππ
+ βu*πβ0,ππ+ βπ βcurlyπβ0,π
+ βπ βfβdiv,π +β
πΉβππβ©Ξ
β1/2πΉ β[πΎπ(f)]πΉ β0,πΉ . (4.24)
Here ππ denotes the union of elements that share a common face with π . Then, it follows from (4.24) and theestimate
βππβfβdiv,ππ
+β
πΉβπ ππβ©Ξ
β1/2πΉ β[πΎπ(f)]πΉ β0,πΉ . β
1/2ππ
(βfβ0,Ξ© +
πβπ=1
βdiv fβ0,Ξ©π+ β[πΎπ(f)]Ξβ0,Ξ
)
that
ππ
(ππ
). β
1/2ππ
C(f) + βy*π β y*ββcurl,Ξ© + βp*π β p*ββcurl,Ξ© + βu*π β u*ββ0,Ξ©
+ βππβcurlyπβ0,ππ
+ βyββcurl,π ππ+ βpββcurl,π ππ
+ βuββ0,π ππ, (4.25)
where C(f) := βfβ0,Ξ© +βπ
π=1βdiv fβ0,Ξ©π + β[πΎπ(f)]Ξβ0,Ξ is well-defined by (2.4). Hence, by the definition of πβ,
we have the estimate for ππ
(ππ
):
ππ
(ππ
). ππ
(ππ
)+ βuπβ0,ππ
, (4.26)
where ππ
(ππ
)has been bounded by (4.25). Since ππ will be marked in the (π + 1)th iteration, it holds that
ππ β T 0π and |πππ
| . ββπβ3β,Ξ©0π
β 0 as π ββ,
by (3.5) and the uniform convergence (4.4) of adaptive meshes. Taking advantage of the absolute continuity ofthe Lebesgue integral and Theorem 4.5, we can get the desired vanishing limit of
ππ
(ππ
)from (4.25) and
(4.26) when π tends to infinity. The proof is complete.
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2035
We are now well-prepared to show that the limiting state y*β and adjoint state p*β actually satisfy thevariational problems (2.7a) and (2.7b), respectively. It is worth emphasizing that in our proof, there is noneed to introduce a buffer layer of elements between the meshes at different levels as in [61], by virtue of thegeneralized convergence result of mesh-size functions (4.4).
Lemma 4.7. Suppose that (y*β,p*β,u*β) is the solution to the limiting optimality system (4.20a)β(4.20c). Thenit satisfies the variational problems (2.7a) and (2.7b), namely,
π΅(y*β,π) = (f + u*β,π)0,Ξ© β π β H0(curl, Ξ©), (4.27)
π΅(p*β,π) =(curly*β β yπ, curlπ
)0,Ξ©
β π β H0(curl, Ξ©). (4.28)
Proof. We only prove (4.27) since the proof of (4.28) is similar. For this, we first introduce the following residualfunctionals on H0(curl, Ξ©):
β(Β·) := π΅(y*β, Β·)β (f + u*β, Β·)0,Ξ©,
andβπ(Β·) := π΅(y*π, Β·)β (f + u*π, Β·)0,Ξ©, π β₯ 0.
Note that Theorem 4.5 gives us the operator-norm convergence:
limπββ
supπβH0(curl,Ξ©)
|β(π)ββπ(π)|βπβH0(curl,Ξ©)
= 0. (4.29)
Therefore, to prove β is actually a zero functional, it is sufficient to show
limπββ
βπ(π) = 0 β π β Cβπ (Ξ©),
by the operator-norm convergence (4.29) and the density of Cβπ (Ξ©) in H0(curl, Ξ©). Recall the standard interpo-
lation operator Ξ β : Hπ (curl, πΊ) β Vβ(πΊ) for 1/2 + πΏ β€ π β€ 1, πΏ > 0, associated with Nedelecβs edge elements,and the corresponding error estimate (cf. [44], Thm. 5.41):
βvβ Ξ βvβcurl,πΊ . βπ (βvβπ ,πΊ + βcurl vβπ ,πΊ), (4.30)
where πΊ is a polyhedral Lipschitz subdomain of Ξ© and Vβ(πΊ) is the lowest-order conforming edge elementspace without the specified boundary condition. Defining w := πβ Ξ ππ β H0(curl, Ξ©) and applying the quasi-interpolation operator Ξ π introduced in Lemma 3.1 to w, we have, by definition of βπ and Ξ ππ, Ξ πw β Vπ,
βπ(π) = π΅(y*π,π)β (f + u*π,π)0,Ξ© = π΅(y*π,wβΞ βw)β (f + u*π,wβΞ βw)0,Ξ©.
By the splitting of the mesh Tπ introduced in (4.3), a derivation similar to the one for (3.17) and (3.19) gives
|βπ(π)| = |βπ(wβΞ βw)| .β
πβTπ
(π(1)π¦,π + π
(2)π¦,π
)βwβcurl,Ξ©π
+β
πβTπ
βπΉβππβ©Ξ©
(π(1)π¦,πΉ + π
(2)π¦,πΉ
)βwβcurl,Ξ©π
.β
πβT +π
ππ(π )βπβ Ξ ππβ0,Ξ©π+
βπβTπβT +
π
ππ(π )βπβ Ξ ππβ0,Ξ©π, (4.31)
where π is an iteration index less than π (clearly, T +π β Tπ). Note that
ππ(π ) β€ ππ
(ππ
)= max
πβTπ
ππ(π )
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2036 B. LI AND J. ZOU
holds for all π β T +π , where ππ is as defined in Lemma 4.6. We also observe from (4.30) that
βπβ Ξ ππβ0,Ξ©π. βπβπ»π (curl,Ξ©).
Hence, by these two observations and Cauchyβs inequality, (4.31) implies
|βπ(π)| . #(T +π )ππ
(ππ
)βπβπ»π (curl,Ξ©) +
ββ βπβTπβT +
π
π2π(π )
ββ 1/2
βπβ Ξ ππβ0,Ξ©0π. (4.32)
To derive the second term in the right-hand side of (4.32), we have also used another observation, by definitionand the shape regularity of the meshes, that for any π β TπβT +
π , there exists an element π β² β T 0π such that
π β π β² and the extended neighborhood Ξ©π of π in Tπ is contained in the extended neighborhood Ξ©π β² of π β² inTπ, i.e., Ξ©π β Ξ©π β² , which, along with the properties (3.6), allows us to writeβ
πβTπβT +π
βπβ Ξ ππβ20,Ξ©π=
βπ β²βT 0
π
βπβTπ,πβπ β²
βπβ Ξ ππβ20,Ξ©πβ€ πΆ
βπ β²βT 0
π
βπβ Ξ ππβ20,Ξ©π β²,
with πΆ independent of the meshes. For a fixed π, by Lemma 4.6, the first term in (4.32) vanishes when π tendsto infinity. For the second term in (4.32), we note from the inequality (4.24) and the boundedness of numericalsolutions (y*π,p*π,u*π) that β
πβTπβT +π
π2π(π ) β€
βπβTπ
π2π(π ) β€ πΆ, (4.33)
where the constant πΆ is independent of π. We hence have from (4.32) that
lim supπββ
|βπ(π)| . lim supπββ
βπβ Ξ ππβ0,Ξ©0π. (4.34)
We note that Ξ©0π inherits a triangulation from the mesh Tπ in the sense that Ξ©0
π =β
πβTπ,πβΞ©0ππ . Then the
interpolation error estimate (4.30) gives
βπβ Ξ ππβ0,Ξ©0π. ββπβπ
β,Ξ©0π
βπβπ»π (curl,Ξ©),
which, by (4.34), implies
lim supπββ
|βπ(π)| . ββπβπ β,Ξ©0
π
βπβπ»π (curl,Ξ©). (4.35)
Letting π ββ in (4.35) and recalling the uniform convergence of mesh funcitons (4.4), we can readily concludethat for all π β Cβ
π (Ξ©), there holdsβ(π) = lim
πβββπ(π) = 0,
which completes the proof.
Lemma 4.7 suggests that it suffices to show that u*β is actually the minimizer to the cost functional π½(u)over the admissible set Uππ. For this, we prove the following result by exploiting the similar splitting of Tπ asin Lemma 4.7 and some fundamental properties of πΏ2-projections.
Theorem 4.8. Under the same assumptions as in Lemma 4.7, u*β satisfies the variational inequality:(p*β + πΌ
(u*β β uπ
),uβ u*β
)0,Ξ©
β₯ 0 β u β Uππ. (4.36)
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2037
Proof. Note that the convex set D+(Ξ©) := v β Cβπ (Ξ©) | v β₯ 0 is dense in Uππ with respect to the πΏ2-norm.
It suffices to show that the variational inequality (4.36) holds for all u in D+(Ξ©). However, we have(p*β + πΌ
(u*β β uπ
),uβ u*β
)0,Ξ©
β₯ 0 β u β Uππβ ,
since (y*β,p*β,u*β) is the solution to the limiting optimality system (4.20a)β(4.20c). The above fact, along withthe relation Uππ
π β Uππβ , implies that for all u β D+(Ξ©), there holds(
p*β + πΌ(u*β β uπ
),uβ Pπu + Pπuβ u*β
)0,Ξ©
β₯(p*β + πΌ
(u*β β uπ
),uβ Pπu
)0,Ξ©
. (4.37)
It follows from the strong convergence of (y*π,p*π,u*π) and βuβ Pπuβ0,Ξ© β€ 2βuβ0,Ξ© that
lim infπββ
(p*β + πΌ
(u*β β uπ
),uβ Pπu
)0,Ξ©
= lim infπββ
(p*π + πΌ
(u*π β uπ
),uβ Pπu
)0,Ξ©
. (4.38)
Noting that Iβ Pπ is an orthogonal πΏ2-projection and using the Poincare inequality for (Iβ Pπ)u:
β(Iβ Pπ)uβ0,π . βπ ββuβ0,π ,
we derive(p*π + πΌ
(u*π β uπ
),uβ Pπu
)0,Ξ©
=(
(Iβ Pπ)(p*π β πΌuπ
), (Iβ Pπ)u
)0,Ξ©
.β
πβTπ
(βπ βp*π β Pπp
*πβ0,π + βπ βuπ β Pπu
πβ0,π
)ββuβ0,π
.β
πβT +π
βπ Β· ππ(π )ββuβ0,π +β
πβTπβT +π
βπ Β· ππ(π )ββuβ0,π , (4.39)
by the definition of ππ(π ). Here we have used the same splitting of Tπ as the one in Lemma 4.7 with π beinga fixed iteration index less than π. Similarly to the proof of Lemma 4.7, we estimate the two terms in (4.39),respectively, as follows. For the first term in (4.39), we haveβ
πβT +π
βπ Β· ππ(π )ββuβ0,π . #(T +π )ππ
(ππ
)ββuβ0,Ξ©,
which, by Lemma 4.6, vanishes as π ββ. For the second term in (4.39), note from (4.33) thatβπβTπβT +
π
π2π(π ) =
βπβTπβT +
π
π2π(π ) + osc2
π
(uπ)β€ πΆ
holds with the constant πΆ independent of π. Then, by Cauchyβs inequality, we haveβπβTπβT +
π
βπ Β· ππ(π )ββuβ0,π . ββπββ,Ξ©0πββuβ0,Ξ©,
which vanishes when π tends to infinity. By above arguments, we have proven
limπββ
|(p*π + πΌ(u*π β uπ),uβ Pπu)0,Ξ©| = 0. (4.40)
In view of (4.37), (4.38) and (4.40), we complete the proof, since the left-hand side of (4.37) is independent ofπ.
We end our theoretical analysis for the adaptive approximation of the control problem with an additionalremark.Remark 4.9. If the function uπ has the regularity: uπ β H1(Ξ©), then using Theorem 4.1, the uniform conver-gence (4.4) and the Poincare inequality, we can argue in a manner similar to Theorem 6.2 of [61] to concludethe convergence of the error estimators ππ. However, for a general πΏ2-data uπ, the optimal control u* canonly be guaranteed to have a πΏ2-regularity, and the data oscillation oscπ
(uπ) may not converge to zero in the
adaptive process.
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2038 B. LI AND J. ZOU
5. Numerical experiments
In this section, we provide a detailed documentation of the numerical results to illustrate the performanceof our adaptive algorithm for the optimal control problem. All the examples presented below are implementedbased on the MATLAB package iFEM [18] using MATLAB 2017a on a personal laptop with 8.00 GB RAM anddual-core 2.4 GHz CPU. When solving the discrete optimization problems (2.12) and (2.13), we use the projectedgradient algorithm in which the HX-preconditioner [32] is also adopted for solving the H(curl)-elliptic problem.More precisely, we start with an initial guess u(0)
β β Uππβ , and in πth step (π β₯ 1) we solve two H(curl)-elliptic
problems to obtain the state y(π)
β and the adjoint state p(π)
β with the help of HX-preconditioner. Then the controlvariable uβ can be explicitly updated as follows:
u(π)β = max
0,u
(πβ1)β + π
(βPβp
(π)β β πΌ
(u
(πβ1)β β uπ
β
)), (5.1)
where the parameter π can be computed explicitly by solving a one dimensional quadratic optimization problem.Compared to the algorithm in [35] where the edge element was used to approximate the control and an additionalleast squares problems with inequality constraints need to be solved to realize the box constraint (1.2), whichis very expensive and time-consuming, our algorithm uses only about half the number of DoFs for computingthe control variable but gives equally accurate numerical resolution, and can be trivially implemented.
In the following, we carry out some numerical experiments for two benchmark problems. For both examples,we shall first compute the numerical solutions and errors on the uniform meshes with five refinement iterationsfor the purpose of comparison, and then conduct the new adaptive algorithm, which is terminated when thecorresponding DoFs are almost the same as that of the finest uniform mesh. The first example is modifiedfrom [22, 67], where there is a corner singularity in the solution. We consider an optimal control problem onthe L-shape domain: Ξ© = [β1, 1]3β[0, 1]Γ [0, 1]Γ [β1, 1] with both the electric permittivity π and the magneticpermeability π taken to be 1. We set the target applied current density (control) uπ and the target magnetic field(state) yπ to zero, and set the parameter πΌ to 0.1. We choose the inhomogeneous Dirichlet boundary conditionand the source term f such that the exact solution are given as follows:
u* = p* = 0 and y* = grad(
π23 sin
(23π
))in cylindrical coordinates.
We set π = 0.5 in the marking strategy (3.4) and remark that there is no need to consider the data oscillationoscπ
(uπ)
in this example because of the vanishing desired control uπ.In our simulations, the computation starts on a very coarse mesh with 480 DoFs. The adaptively refined
meshes in the 10th, 14th and 18th iterations are presented in Figure 1, in which we clearly observe that themeshes are locally refined around the π§-axis where the singularity of y* occurs. Note the non-smooth functiony* /β H1(Ξ©) and that the lowest-order edge elements of first family are used to discretize the state variable.We typically cannot expect the optimal convergence order π(β) on a uniformly refined mesh according to thestandard interpolation result (4.30). However, this theoretical order of convergence can be recovered by theadaptive mesh refinement in the sense that
βy*π β y*βcurl,Ξ© . (#DoFs)β1/3,
which has been confirmed by the convergence history plotted in Figure 2 (left) on a double logarithmic scale. Weshould also observe that the errors of the control βu*βu*πβ0,Ξ© and the adjoint state βp*βp*πβcurl,Ξ© reduce witha slope β2/3, which is twice the one of the error associated with the state y. The difference in the convergencerates may be explained by the fact that u* and p* are zero functions and hence very smooth while the optimalstate y* is non-smooth and has a strong singularity near the π§-axis. Moreover, Figure 2 (right) shows thereduction of the total error (almost dominated by the error βy* β y*πβcurl,Ξ©):
βy* β y*πβcurl,Ξ© + βp* β p*πβcurl,Ξ© + βu* β u*πβ0,Ξ©,
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2039
Figure 1. Evolution of the adaptive mesh for the first example in steps π = 10, 14 and 18(final mesh).
Figure 2. Convergence histories of the control, state and adjoint state (left), and the totalerrors for the uniform mesh refinement and the adaptive mesh refinement, as well as the errorestimator πβ (right) for the first example.
and the convergence behavior of the a posteriori error estimator πβ = πβ, which confirms the efficiency of πβ
and the superiority of the adaptive mesh refinement over the uniform mesh refinement: the total error on theadaptively refined mesh reduces with an order β1/3, double the one (β0.15) on the uniformly refined mesh.This is also confirmed by the computing times and computational costs: for achieving the error over the meshgenerated by the 5th uniform refinement, the adaptive algorithm takes only about 80 s and 105 DoFs, whereasthe uniform one takes about 580 s and 1 410 240 DoFs.
The second example is chosen to be similar to the one in [35], where there are non-smooth source terms andlarge jumps of physical coefficients across the interface between two different media. To be precise, we consideran optimal control problem on Ξ© = [β1, 1]3 with a high-contrast inclusion: Ξ©π := x β R3; π₯2 +π¦2 +π§2 < 0.62,
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2040 B. LI AND J. ZOU
Figure 3. Evolution of the adaptively refined mesh (2D slice) for the second example in stepsπ = 8, 12, 16 and 19 (final mesh).
on which the coefficients π and π are given by
π =
10 in Ξ©π,
1 in Ξ©βΞ©π,πβ1 =
0.1 in Ξ©π,
1 in Ξ©βΞ©π.
We set the target state yπ = 0 and the target control uπ = 10(πΞ©π , 0, 0). Here πΞ©π is the characteristic functionof Ξ©π. We define a scalar function:
π(x) =1
2πsin(2ππ₯) sin(2ππ¦) sin(2ππ§),
and then introduce the non-smooth source term f:
f(x) = πβπ(x)β 10(πΞ©π, 0, 0).
We readily see that the unique solution to the optimality system (2.7a)β(2.7c) is given by
y* = βπ, p* = 0, and u* = 10(πΞ©π, 0, 0).
Other parameters are chosen to be same as the first example.We still set π = 0.5 in the marking strategy (3.4) and start our computation on a very coarse initial mesh
with 604 DoFs. We show the evolution of adaptive meshes on the (π¦, π§)-cross section at π₯ = 0 in Figure 3,from which we immediately see that the meshes are strongly concentrated in the high-contrast inclusion Ξ©π andclearly capture the shape of the interface. We can also observe, from Figure 4 (left), an optimal convergenceorder β1/3 for the control and state variable and a little bit faster decrease of the error for the adjoint stateβp*βp*πβcurl,Ξ©, which is possibly because p* is quite smooth (a zero function), compared to u* and y*. Figure 4(right) again verifies the effectiveness of the error estimator ππ and the gain of computational efficiency from theadaptive mesh refinement: the total error on the adaptively refined mesh reduces with an order β1/3, whereasthe error on the uniform one reduces only with an order β0.2.
6. Concluding remarks
In this work, we have studied an electromagnetic optimal control problem and used the lowest-order edgeelements to approximate the state and adjoint state, while used the piecewise constant functions to approxi-mate the control. We have designed an adaptive finite element method with an error indicator involving boththe residual-type error estimator and the lower-order data oscillation. We have established the reliability andefficiency of the a posteriori error estimator and the strong convergence of the adaptive finite element solutions
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AN ADAPTIVE EDGE ELEMENT METHOD AND ITS CONVERGENCE 2041
Figure 4. Convergence histories of the control, state and adjoint state (left), and the totalerrors for the uniform mesh refinement and the adaptive mesh refinement, as well as the errorestimator πβ (right) for the second example.
for both the state and control variables. With a very minor modification, our analysis and arguments can bedirectly applied to the more realistic case [61, 63] where the control u is added on a subdomain Ξ©π of Ξ© andthe case [27] where the control satisfies a bilateral constraint. It is worth pointing out that our arguments canalso be modified to cope with the inhomogeneous Dirichlet boundary condition. To be exact, suppose that theboundary condition is given by πΎπ‘y = πΎπ‘g with g β H(curl, Ξ©) being a known function satisfying the givenDirichlet trace data [14]. Then we can check that the optimality systems (2.7a)β(2.7c) and (2.14a)β(2.14c) stillhold, except that (2.7a) and (2.14a) are solved on the affine spaces g + V and Ξ βg + Vβ, respectively. Hence,up to a possible data oscillation: βgβ Ξ βgβcurl,Ξ©, our a posteriori error estimates and the convergence analysiscan be readily applied.
As mentioned in the Introduction, the model of our interest can be connected with the discretization of thecontrol problem of time-dependent eddy current equations [41,46,47], where the implicit time-stepping schemeis adopted for the sake of stability [10,31]. In this case, the coefficient π will be scaled by the current time-stepsize. Therefore, it is important to design an error estimator which is robust with respect to the scaling of thecoefficients, namely, the generic constants involved in the a posteriori error analysis should be independentof the scaling factors of the coefficients. For this, we may measure the errors of the states (y and p) by theenergy norm on H0(curl, Ξ©): βΒ·β2π΅ = β
βπβ1curlΒ·β20,Ξ© + β
βπΒ·β20,Ξ©, and the error estimators may also need to
be scaled correspondingly. If we assume that π and π are element-wise constant with respect to the initialmesh T0 [10], or that there exists a scaled norm β Β· β2V := πβ1
* βcurl Β· β20,Ξ© + π*β Β· β20,Ξ© with π* and π* beingpositive constants, such that π΅(Β·, Β·) is continuous and inf-sup stable with respect to β Β· βV with the involvedgeneric constants independent of π and π [53], one can naturally follow the a posteriori error analysis providedin this work to obtain a robust error analysis by using the energy norm and the scaled error estimators (forinstance, if the norm βΒ·βV defined above exists, by adding a scaling factor
βπ*, we can modify π(1)
π¦,π and π(1)
π¦,πΉ
as follows: π(1)
π¦,π := βπβ
π*βf + u*β β curlπβ1curly*β β πy*ββ0,π and π(1)
π¦,πΉ :=β
βπΉ π*β[πΎπ‘(πβ1curly*β)]πΉ β0,πΉ ).We refer the readers to [17, 48] for related discussions. However, the detailed and rigorous treatments for thegeneral coefficients and the time-dependent model are not trivial tasks and need further investigations. Wefinally remark that it is also of great interest to design the adaptive algorithm for more complicated models,such as the nonlinear electromagnetic control problem [64] and the Maxwell variational inequalities [65,66].
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2042 B. LI AND J. ZOU
Acknowledgements. The work of Jun Zou is substantially supported by Hong Kong Research Grants Council (projects14306718 and 14306719). The authors would like to thank the anonymous referees for their very careful reading of themanuscript and their numerous constructive comments and suggestions, which have helped us improve the presentationand results of this work essentially.
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