Analysis of 2D Maxwell’s Equations in a Time-Harmonic Regime

The variational formulation is an essential tool to analyze the existence and uniqueness of the solution of certain partial di ﬀ erential equations with boundary conditions. We can further approximate this analytical solution by computing a corresponding numerical solution obtained by the ﬁnite element method. In this paper, we studied 2D Maxwell’s equations in a time-harmonic regime. We established a corresponding variational formulation and proved its well-posedness in certain conditions. We also constructed a corresponding internal approximation and gave an error estimate within some prior assumptions. This theoretical analysis provides a basis to compute the numerical solution of time-harmonic 2D Maxwell’s equations and gives physical signiﬁcance to the transverse magnetic problem.


Introduction
In mathematics, more precisely in differential calculus, a partial differential equation (sometimes abbreviated as PDE) is a differential equation that has unknown functions as solutions; these functions depend on several variables that satisfy certain conditions concerning their partial derivatives. (Evans, L. C., 2010;Pinchover, Y. & Rubinstein, J., 2005) It is hard to analyze the entire set of solutions of a PDE problem, but the boundary conditions often reduce the set of solutions down to a few. Unlike the parameters of the solution sets of an ordinary differential equation, which correspond to the additional conditions, the boundary conditions for PDEs are instead in the form of a function; intuitively, this indicates that it is much more difficult to analyze the solution set, which is true in almost all problems.
The variational point of view enables us to approach certain problems of partial differential equations from an unusual perspective that is rich and powerful. In particular, it allows us to introduce the theoretical elements leading to the solution of the problem (proving the existence and uniqueness of the solution in an adequate framework) and build the finite element method, which depends on the theoretical considerations in order to naturally provide a method to approach the solution (which, very often, is not otherwise explicitly computable).
In the preliminaries, we introduce the variational formulation and Sobolev spaces, which are essential to give the wellposedness of PDE problems. In fact, we can transform PDE problems with boundary conditions into variational formulation problems using Greens formula; then, we can study the corresponding variational formulations. We also introduce an important theorem, the Lax-Milgram theorem, which gives the well-posedness of some variational formulations, and we use it to prove our further research in section 3. In addition, we introduce the finite element method, which is a numerical method that computes solutions of certain boundary problems. The principle of the finite element method is to substitute the Hilbert space V, on which the variational formulation is posed, with a finite dimensional subspace V h . The internal approximation posed on V h can be reduced to a simple resolution of a linear system. Moreover, one can carefully construct a V h so that it accurately approximates V and the solution u h in V h is "close" to the actual solution u in V.
Though prior studies have analyzed other Maxwell's equations and curl problems (Ciarlet, Wu & Zou, 2014;, the investigation of the transverse magnetic problem within the transverse mode of electromagnetic radiation is lacking. Our current research studies the 2D Maxwells equations in a time-harmonic regime that models this problem. By using Greens formula and some analysis, we established a corresponding variational formulation. Then, we studied the well-posedness characteristics of this variational formulation by discussing the different cases of the coefficient ω in the variational formulation. We found that the variational formulation is well-posed when ω is not a real number, and also when ω is a real number and the homogeneous variational formulation has a unique solution. But when ω is a real number and the homogeneous variational formulation has a non-zero solution, the variational formulation may not be well-posed, and we will analyze two situations here. For the numerical approach, we established a corresponding internal approximation, which can be well characterized, to easily obtain its well-posedness. Finally, we adapted some conclusions to derive an estimate for the rate of convergence between the solution of the corresponding internal approximation and that of the variational formulation within some prior assumptions.

Preliminaries
In this paper, we explore within the space Ω, which is the domain of R N , and we denote ∂Ω its boundary. However, we sometimes assume that Ω is a regular bounded domain, one that locates on only one side of its regular hypersurface boundary. We denote n the unit vector normal to ∂Ω oriented to the external of Ω. Moreover, we denote dx the volume measure in Ω and ds the surface measure on ∂Ω.

Hilbert Space
We first introduce the definition for a Hilbert space.
Definition 2.1. The vector space V equipped with the inner product ·, · V is a Hilbert space if it is complete for the norm (2.1) A Hilbert space has the following characteristics, which will be frequently used in other subsections.
We note in this case v n v ∞ when n tends to +∞.
In fact, if {v n } n∈N in V converges strongly to v ∞ ∈ V, then {v n } n∈N converges weakly to v ∞ ∈ V, and this can be proved using Cauchy-Schwarz inequality.
We also have the following theorem that indicates the relationship between weak convergence and boundedness.
Theorem 2.3. If {v n } n∈N is a bounded sequence in V, then we can extract a sub-sequence {v n } that converges weakly. Conversely, if {v n } n∈N is a sequence that converges weakly in V, then {v n } is bounded.
Another way to prove that a space is a Hilbert space is using the following lemma: Lemma 2.4. Any closed subspace of a Hilbert space is also a Hilbert space.

Lebesgue Space
Another common function space is Lebesgue space.
In this paper, we mainly consider functions in the space L 2 (Ω). Moreover, we admit that L 2 (Ω) is a Hilbert space equipped with the inner product u, v L 2 (Ω) = Ω uvdx.

Sobolev Space
Before introducing Sobolev spaces, we have the following definitions: Definition 2.6. A function u in L 2 (Ω) is weakly differentiable if there exist functions (w i ) 1≤i≤N ∈ L 2 (Ω) such that for any function ϕ ∈ C ∞ c (Ω), we have We call w i the i-th weak partial derivative of u, denoted as ∂u ∂x i .
where ∂u ∂x i is the weak partial derivative of u. More generally, H p (Ω), for p ≥ 2, is defined by: Proposition 2.9. The Sobolev space H p (Ω) with the inner product is a Hilbert space.
Proof. Proving for p > 1 is similar to that for p = 1, so it suffices to prove for the case p = 1. We admit that L 2 (Ω) is a Hilbert space. As the inner product is defined, we need to prove that H 1 (Ω) is complete. We assume that the sequence {u n } n∈N is a Cauchy sequence in H 1 (Ω). By the equation there exists Cauchy sequences {u n } n∈N and { ∂u n ∂x i } n∈N which converge to u and ω in L 2 (Ω) respectively. From Definition 2.6, for any ϕ ∈ C ∞ c (Ω), we have: (2.4) As the limit n → ∞, we obtain that ∂u ∂x i = ω. Therefore, u n converges in H 1 (Ω).
Lemma 2.10. Let Ω be a regular bounded open set of class C 1 , then C ∞ c (Ω) is dense in H 1 (Ω) and H 2 (Ω).
To evaluate the boundary problems later in this paper, we will define the "edge value", or the "trace" of v on the ∂Ω, of a function in H 1 (Ω) through the following theorems.
Theorem 2.11. (Trace Theorem for H 1 (Ω)) Let Ω be a regular bounded open set of class C 1 . The trace application T 0 is defined by: This application T 0 extends by continuity into a continuous linear application from H 1 (Ω) into L 2 (∂Ω), also denoted as T 0 . Specifically, there exists a constant C > 0 such that for any function u ∈ H 1 (Ω), Theorem 2.12. (Trace Theorem for H 2 (Ω)) Let Ω be a regular bounded open set of class C 1 . The trace application T 1 is defined by: with ∂v ∂n = ∇u · n. This application T 1 extends by continuity into a continuous linear application from H 2 (Ω) to L 2 (Ω). Specifically, there exists a constant C > 0 such that for any function u ∈ H 2 (Ω), By Definition 2.10 and these trace theorems, we deduce Green's formula below.
Theorem 2.13. (Green's Formula) Let Ω be a regular and bounded domain. If u and v are functions of H 1 (Ω), then we have: Moreover, if u ∈ H 2 (Ω) and v ∈ H 1 (Ω), then we have: Remark 2.14. Since C ∞ c (Ω) is dense in H 1 (Ω) and H 2 (Ω), we first construct two sequences {u n } and {v n } in C ∞ c (Ω) which converge to u ∈ H 1 (Ω) and v ∈ H 1 (Ω) respectively. We notice that u n and v n satisfy (2.5), and as n → +∞, we can deduce (2.5). The derivation of (2.6) is similar.
To deduce the equivalence of certain variational formulations later in the paper, we introduce the following lemma.

Lax-Milgram theorem and Finite Element Method
To evaluate the well-posedness of a partial differential equation with certain boundary conditions, we follow a general approach. We first abandon the space C k (Ω) of continuously differentiable functions in favor of its "generalization", a Hilbert space V. Then, we multiply a test function v ∈ V to both sides of the equation and integrate within Ω. Moreover, we reduce the order of the equation by one using Green's formula and arrive at a corresponding variational formulation, sometimes known as the weak solution of the original partial differential equation. Next, we evaluate the well-posedness of the variational formulation through the Lax-Milgram theorem. Finally, we use the finite element method to approximate for a numerical solution with certain estimate of error.
Before investigating a specific partial differential equation, we need to briefly introduce the Lax-Milgram theorem and the finite element method. In this paper, we consider variational formulations in the form of The assumptions on a(·, ·) and L(·) are: (1) a(·, ·) is a continuous bilinear form on V, i.e., there exists M > 0 such that: (2.9) (2) L(·) is a continuous linear form on V, i.e., there exists C > 0 such that: (2.10) (3) a(·, ·) is coercive, i.e., there exists T > 0 such that: (2.11) Theorem 2.16. (Lax-Milgram Theorem) Let V be a real Hilbert space. If L(·) is a continuous linear form on V and a(·, ·) is a continuous and coercive bilinear form on V, then the variational formulation in the form of (2.8) admits a unique solution.
After analyzing the existence and uniqueness of the solution, we wish to further approximate this solution through a numerical approach -the finite element method. The fundamental idea of this approach is derived from the variational approach introduced previously, where we evaluate within a finite dimensional subspace of V, V h , reduce the internal approximation to a linear system of matrix, and solve for its solution numerically. Furthermore, to determine the estimated error of the numerical solution, we measure the accuracy of u h as an approximation of u.
We use the same abstract framework of the variational formulation in (2.8). However, to compute a numerical solution, we use the finite dimensional subspace V h to replace the Hilbert space V.
To guarantee the existence and uniqueness of a numerical solution to the internal approximation (2.12), we have the following lemma.
Lemma 2.17. Let V be a Hilbert space and V h be its finite dimensional subspace. If L(v) is a continuous linear form on V and a(u, v) is a continuous and coercive bilinear form on V, then the internal approximation (2.12) admits a unique solution that can be computed through solving a linear system of equations with positive definite matrix.
Proof. Let V h be a finite dimensional space with (φ j ) 1≤ j≤N h as its finite basis. We consider v h = φ i and u h = N h j=1 u j φ j . Then (2.12) is equal to: To express as a linear system, we define the following matrixes: Now, for 1 ≤ i, j ≤ N h , the internal approximation becomes: The coercivity of the bilinear form a(u, v) suggests the positive definiteness of (K h ), which means for any vector U h ∈ R N h , we have Since K h is invertible, the matrix problem (2.13) admits a unique solution.
Before introducing the finite element method P k in a d-dimensional space(d ≥ 2), where P k denotes a k-th order polynomial, we give the definition of the triangulation of a polyhedral domain.
Definition 2.18. Say that Ω is a connected and open polyhedral of R d . We call the set T h of d-simplexes (κ i ) 1≤i≤n a triangulation of Ω if it verifies: (1) κ i ⊂Ω such thatΩ = ∪ n i=1 κi.
(2) For two distinct d-simplexes κ i and κ j , their intersection forms a N-simplexes (0 ≤ N ≤ d − 1), whose vertices correspond to those of κ i and κ j .
The T h mesh has vertices that correspond to those of the d-simplexes κ i that compose it. Moreover, h denotes the maximum diameters of the d-simplexes κ i .
We observe that the finite element method P k is only applicable to a polyhedral domain. Then, we give the definition of finite element method P k : Definition 2.19. For a mesh T h of a connected and open polyhedral domain Ω, we define the finite element method P k by the following space: (2.14)

Establishment of the Variational Formulation
Let Ω ⊂ R 2 be a connected bounded domain with a regular (class C 1 ) connected ∂Ω boundary. We denote n = (n x , n y ) the exterior normal to ∂Ω. We introduce real-valued functions ω the frequency of the time-harmonic field and ε, µ ∈ C 1 (Ω) the dielectric permittivity and magnetic permeability of the medium. We will assume that ε −1 , µ −1 ∈ C 1 (Ω) so that there exist constants α > 0, β > 0 such that The Maxwell problem that we want to study is the transverse magnetic problem, which governs the relationship between the electric field strength u and the current density J on a propagation plane. Mathematically, we only consider this relationship in a single period, so we algebraically cancelled the exponential function e iωt which shows how the variables change periodically and harmonically with time. The resultant partial differential equation is written below: (Ω) and curlϕ = ∂ϕ ∂y , − ∂ϕ ∂x . In this topic, unless otherwise stated, the functions will be complex-valued. Despite ω is real-valued, we still consider the case of ω ∈ C\R for its easily-derived well-posedness but puts a focus on the case when ω ∈ R. In the interest of concision, we avoid the obvious case of ω = 0.
We will need the following (complex) version of the Lax-Milgram theorem: 1. Let X be a Hilbert space of complex-valued functions.
2. Let l(·) be an antilinear form continuous on X.
We introduce two functional spaces that will serve in the analysis below. For ϕ ∈ C 1 (Ω) and Corollary 3.1. By Green's formula, we can establish the identity We will then note curl v = V. With (3.3), we can verify that this definition extends the one given in (3.2) for regular functions.
Let us introduce the spaces: where the operator curl is understood in the weak sense according to the previous definition. We will admit that (3.3) is still right for all ϕ ∈ H 1 (Ω), v ∈ H(curl; Ω).
Proof. With the inner products defined as above, it remains to prove that H(curl; Ω) and V T (µ; Ω) are complete. We first establish the identity: u 2 H(curl;Ω) = curl u 2 L 2 (Ω) + u 2 L 2 (Ω) . We assume that { u n } n∈N is a Cauchy sequence in H(curl; Ω). Since L 2 (Ω) is complete, { u n } and {curl u n } are both Cauchy sequences that converge in L 2 (Ω). We note that there exists limits u ∈ L 2 (Ω) such that u n → u in L 2 (Ω) and ω ∈ L 2 (Ω) such that curl u n → ω in L 2 (Ω). To further prove that curl u n → curl u, we first use (3.3) to obtain Ω curlϕ · u n dxdy = Ω ϕ curl u n dxdy, ∀ϕ ∈ C ∞ c (Ω).
Based on Definition 3.2, we note that ω = curl u. Since curl u n → curl u in L 2 (Ω) and u n → u in L 2 (Ω), we obtain that u n → u in H(curl; Ω) and H(curl; Ω) is a Hilbert space.
The Maxwell problem that we want to study is: where J denotes a source term belonging to C 1 (Ω).
Before applying the complex Lax-Milgram theorem to this Maxwell problem, we first establish its variational formulation.
Using the identity (3.3), we have Since C ∞ c (Ω) belongs to H 1 (Ω), we let v ∈ C ∞ c (Ω) and we deduce that Now, the expression coincides with the one in Lemma 2.15: And therefore, by Lemma 2.15, we have curl ε −1 curl u − ω 2 µ u = curl ε −1 J in Ω.
We assume that the image of the application ϕ → ϕ · τ defined on H 1 (Ω) × H 1 (Ω) is dense in L 2 (∂Ω). As the partial differential equations are equivalent in Ω, we only evaluate the equations on ∂Ω, which is Since the image of the application v → v· τ defined on H 1 (Ω)×H 1 (Ω) is dense in L 2 (∂Ω), we deduce that ε −1 (curl u−J) = 0 on ∂Ω.
Remark 3.6. However, analyzing the well-posedness characteristics of the variational formulation posed in H(curl; Ω) is less than satisfactory as the injection of H(curl; Ω) into L 2 (Ω) is not compact, thereby we cannot extract sub-sequences in H(curl; Ω) that converge strongly and weakly in L 2 (Ω). Therefore, we want to establish an equivalent variational formulation posed in V T (µ; Ω). To do so, we need to introduce the Rellich-Kondrachov theorem for H(curl; Ω) and Poincar inequality for H 1 # (Ω), a new space defined by H 1 # (Ω) := ϕ ∈ H 1 (Ω) | Ω ϕdxdy = 0 .
Then we recall a conclusion in , which gives a compactness embedding for V T (µ; Ω) into L 2 (Ω).
We know that {v n } ∈ H 1 # (Ω) is bounded because we have By Lemma 3.7, we can find a sub-sequence {v n } which converges in L 2 (Ω). Observing that ∇v n 2 L 2 (Ω) ≤ 1 n , we can deduce that ∇v n L 2 (Ω) → 0 as n → ∞.
Then we have the following proposition.
Proposition 3.9. For v ∈ H(curl; Ω) given, the problem admits a unique solution.

Well-Posedness Characteristics
After arriving to the variational formulation (3.9), the weak solution of the Maxwell problem (3.5), we can analyze the existence and uniqueness of the solution u. We do so under the following cases: 2. Find u ∈ V T (µ; Ω) for ω ∈ R, i when P 0 has a unique solution.
ii when P 0 has a non-zero solution.
In the above, P 0 denotes the problem (3.9) for ω 0 ∈ R and (·) = 0: (3.10) We begin by considering the first case and introducing its corresponding result.
Subsequently, we consider the two sub-cases of the second case and introduce their results.
Proof. From Theorem 3.11, we denote u δ the unique solution of (3.9) for ω δ = ω 0 + iδ and δ > 0. We wish to prove the existence of a constant C > 0 independent of δ such that u δ H(curl;Ω) ≤ C, ∀δ ∈ 0, 1 . (3.11) To do this, we will reason by the absurd by assuming that there is a sequence {δ n } such that lim n→+∞ δ n = 0 and u δ n H(curl;Ω) > n. We define w n = u δ n / u δ n H(curl;Ω) .
Here { u δ n } is a sequence of the solution to (3.9) with ω δ n = ω 0 + iδ. Substituting { u δ n } and {ω δ n } into (3.9), we have Observing that u δ n = w n · u δ n H(curl;Ω) , we deduce that As n → ∞, we have ω δ n → ω 0 and u δ n −1 H(curl;Ω) → 0, we obtain the following equation: (3.14) In addition, we notice that { w n } is bounded as w n H(curl;Ω) = u δn H(curl;Ω) u δn H(curl;Ω) = 1. From Lemma 3.7, we can find a subsequence of { w n } that is convergent in L 2 (Ω). Similarly, from Lemma 2.3, we can find another sub-sequence which is weakly convergent in V T (µ; Ω). Because they are both sub-sequences of { w n }, we denote them as { w n } for simplicity, where we have w n n→∞ → w in L 2 (Ω) and We use Cauchy-Schwarz inequality to prove that Ω w n¯ vdxdy n→∞ → Ω w¯ vdxdy, and therefore, Ω curl w n curl vdxdy n→∞ → Ω curl wcurl vdxdy.
Therefore, we can simply show that u is a solution of (3.9) with ω = ω 0 ∈ R since Remark 3.13. To discuss the supplement of Theorem 3.12, which is the existence of a solution for (3.9) when ω ∈ R and P 0 has a non-zero solution of u 0 , we consider it in two ways: when ( u 0 ) 0 and when ( u 0 ) = 0.
Proposition 3.14. Suppose that there exists a non-zero solution u 0 for P 0 , then (3.9) does not have a solution when ω ∈ R and u 0 0.
Since u δ H(curl;Ω) ≤ C, we have u δ n → u in L 2 (Ω) and u δ n u in V T (µ; Ω). Using the same deduction as (3.18), we can show that u is a solution of (3.9) with ω = ω 0 ∈ R.

Numerical Approximation
Now, we assume that Ω is a polygon and accept that the previous results remain valid in such a geometry. To study the case ω ∈ R, we would like to work on the formulation (3.9) posed in V T (µ; Ω) so that we can use the fact that the injection of V T (µ; Ω) into L 2 (Ω) is compact. However, it is hard to discretize (3.9) because V T (µ; Ω) contains the integral Ω µ v∇ϕdxdy = 0, which cannot be described by an easily-chosen finite basis φ j (x). Therefore, we need to apply the finite element method in a new space that is possible to interpret geometrically.
Let us define the Hilbert space Y To deduce an equivalent variational formulation posed in Y, we can easily prove that v ∈ H(curl; Ω) belongs to Y if and only if there exists ω ∈ L 2 (Ω) such that Ω µ v · ∇ϕdxdy = Ω wϕdxdy, ∀ϕ ∈ C ∞ c (Ω). (3.21) We consider, for λ > 0, the variational formulation Find u ∈ Y such that (3.22) Proposition 3.16. If u verifies (3.9), then u is a solution of (3.22).

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(3.23) admits a unique solution for any f ∈ L 2 (Ω) if and only if k {k 0 , k 1 , . . .}, where (k n ) is a sequence consist of increasing positive real numbers such that lim n→+∞ k n = +∞.
Proposition 3.17. If u verifies (3.22), then u is a solution of (3.9) when ω 2 λ k n for all n ∈ N. Proof.
Let Y h be a finite dimensional space such that Y h ⊂ Y. We are interested in approximating the solution of (3.22) in Y h , but the conditions on ∂Ω are abstractedly imposed without trying to make explicit of the functions that satisfy it.
For example, we take Y h = ϕ ∈ C(Ω)|ϕ| K i ∈ P k (K i ) , where T h is a trangulation of Ω and P k (K i ) denotes the set of polynomials of degree at most k on the triangle K i .
We define Y h := v ∈ V h × V h | µ v · n = 0 on ∂Ω so we can easily see that Y h ⊂ Y. Then we consider the following discretized problem which will be proven as well-posed.
Proposition 3.18. The discretized problem Find u h ∈ Y h such that for any v h ∈ Y h , (3.24) where ω = iκ with κ ∈ R\{0} for the simplicity of the following, admits a unique solution.
Proof. We verify the condition of the complex Lax-Milgram theorem using Cauchy-Schwarz inequality.
Remark 3.19. Throughout this paper, we do not actually compute a numerical solution to (3.24), but we can still deduce an expected estimate of error for it.
Definition 3.20. To approximate the variational formulation via the form of polynomials, we define an operator r h : where φ j (x) is a finite basis of V h .
Lemma 3.21. (Cea's Lemma) Let V be a Hilbert space and V h be a finite dimensional space in V. If u ∈ V and u h ∈ V h , then Proof. We first construct two variational formulations with u and u h : Observing that V h ⊂ V, we can substitute ω with ω h . By subtracting the two equations, we have |a( u − u h , ω h )| = 0.
When ω h = u h − v h , we can further establish that |a( u − u h , u h − v h )| = 0. By the continuity and coercivity of a( u, v), there exists M > 0 and α > 0 such that |a( u, v)| ≤ M u V v V and a( u, u) ≥ α u 2 V .
Substituting u = u − u h into the coercivity equation, we have Since |a( u − u h , u h − v h )| = 0, we deduce that Reducing the like terms, we obtain that Theorem 3.22. For u ∈ H 2 (Ω) × H 2 (Ω), the estimate of error, which we denote as h, is first-order and we have u − u h Y ≤ Ch u H 2 (Ω) .