Numerical Solution to the Time-Dependent Maxwell Equations in Two-Dimensional
Singular Domains: The Singular Complement Method
F. Assous,∗P. Ciarlet, Jr.,†and J. Segr´e∗
∗CEA/DAM–DIF, BP12-91680, Bruy`eres-le-Chˆatel, France; and†UMA, ENSTA, 32 boulevard Victor, 75739 Paris Cedex 15, France
E-mail: assous@bruyeres.cea.fr
Received August 3, 1999; revised March 6, 2000
In this paper, we present a method to solve numerically the time-dependent Maxwell equations in nonsmooth and nonconvex domains. Indeed, the solution is not of regularity H1 (in space) in general. Moreover, the space of H1-regular fields is not dense in the space of solutions. Thus an H1-conforming Finite Element Method can fail, even with mesh refinement. The situation is different than in the case of the Laplace problem or of the Lam´e system, for which mesh refinement or the addition of conforming singular functions work. To cope with this difficulty, the Singular Complement Method is introduced. This method consists of adding some well-chosen test functions. These functions are derived from the singular solutions of the Laplace problem. Also, the SCM preserves the interesting features of the orig- inal method: easiness of implementation, low memory requirements, small cost in terms of the CPU time. To ascertain its validity, some concrete problems are solved numerically. °c2000 Academic Press
Key Words: Maxwell’s equation; singularities; reentrant corners; conforming finite elements.
1. INTRODUCTION
In recent years, modeling and solving numerically problems which couple charged par- ticles to electromagnetic fields has given rise to challenging mathematical and scientific computing developments. In the industry, a variety of examples can be thought of, such as the ion or electron injectors for particle accelerators, the free electron lasers, or the hyperfre- quency devices. The mathematical model which is most relevant in describing the physics of these devices is the time-dependent coupled Vlasov–Maxwell system of equations.
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In this context, and independently of any geometrical considerations, we have developed a method to solve the time-dependent Maxwell equations (see [6]), which has been de- signed with the help of H(curl)∩H(div)-conforming Finite Elements.1 In this method, the computed electromagnetic field is continuous (this condition is recommended to re- duce the noise of the solution of the coupled problem [7]), and the numerical scheme does not require solving a linear system at each time step. The conditions on the divergence of the fields are treated as constraints and dualized as such by using Lagrange multipli- ers, which yields a saddle-point formulation. We refer the reader to [6] for an in-depth analysis of the method and for a detailed bibliography. This method is also interesting as nodal finite elements (Lagrange P1) are used, the implementation of which is common and easy.
However, a number of industrial structures and objects that must be modeled present a surface with edges, corners, etc., be it intentionally or not. The existence of those geo- metrical distinctive features—of those geometrical singularities—on the boundaries of the domains which must be studied can create singular fields, that is unbounded fields (in the neighborhood of those singularities). Moreover, this difficulty is not restricted to exterior problems, as it must be dealt with in bounded domains too, as soon as the boundary of the domain contains such singularities. Such is the case in the following situations:
• The geometrical singularities are an active part of the device, for instance to generate the powerful electromagnetic fields which are required to extract a strong electron current at a velvet cathode in a microwave generator.
• The geometrical singularities are only the consequence of constraints on the a priori configuration of the device, and it is mandatory to be able to control the induced negative effects, such as the above-mentioned powerful fields which can produce breakdowns.
Finally, let us note that the presence of singularities changes the solution in the whole domain and not only in their neighborhood. In this respect, they have a nonlocal effect. We present an example which illustrates the effect of singularities in a numerical computation.
Here, we focus on the qualitative aspect of the results only.
We consider the propagation of a wave in a stub filter (see Fig. 1), which is propagated or evanescent, depending on both the frequency of the wave and on the size of the stubs. For given frequencies, the guide is transversally closed, thus becoming a plain waveguide of rectangular section. In the example below, the guide is illuminated by a wave, the time-signal of which is included in the range of the frequencies of propagated modes.
This filtering phenomenon can be modeled by the time-dependent Maxwell equations.
The result of a simulation for a two-stub filter is shown in the left-hand side of Fig. 1 for an incident wave with the frequency of a propagated mode. The data and the geometry are assumed to be independent of one of the space variables, and therefore the model is 2D.
Contrary to the physics, the result is that of an evanescent mode for this frequency. In order to emphasize the connection between the odd filtering effect and the reentrant corners (the singularities), these edges are smoothed; that is, they are replaced by smooth structures with a radius in the order of 1/20th of the wavelength. The results that are obtained in this case (see Fig. 1 (right)) confirm that there is indeed a propagated mode. However, the
“smoothing” of the corners yields two problems which must be addressed:
1Thanks to recent results [16], choosing a piecewise smooth H(curl)∩H(div)-conforming FE amounts to using an H1-conforming FE.
FIG. 1. Eycomponent—sharp corners: evanescent mode in the guide—smoothed corners: propagated mode in the guide.
• it requires a mesh refinement in the neighborhood of the reentrant corners (here, by a factor of 10), which leads to high computational costs.
• The original geometry of the device is modified, and the simulation thus loses its accuracy. Moreover, by smoothing the singularites, the very strong fields are numerically underestimated, whereas one of the goals of the simulation is to control the negative effects (breakdowns, etc.) by estimating them precisely.
Let us note however that there is no problem (in this configuration) as far as the Bz
magnetic induction is concerned, either for the original geometry or for the smoothed geometry (cf. Fig. 2). In this respect, Bzis sufficiently smooth.
In order to address the problems raised by the singularities, we provided a mathematical analysis of the singularities of Maxwell’s equations in a nonconvex geometry [5]. The key point is the following: in a nonconvex polygonal (or polyhedral in 3D) domainÄ, the
FIG. 2. Smooth Bzcomponent, the mode is propagated in both cases.
solutions (electric field and magnetic induction) of Maxwell’s equations at a given time t do not belong to H1(Ä)2 (or H1(Ä)3)as is the case in a convex domain, but only to H(curl, Ä)∩H(div, Ä). In other words, the “physical” solution of Maxwell’s equations, which only belongs to the functional space H(curl, Ä)∩H(div, Ä), goes to infinity when one comes close to a reentrant edge or a reentrant corner. It does not coincide with the
“smoothed” solution, that is, the one computed thanks to a formulation of the problem in H1(Ä)2. In addition, there is no hope to converge to the “physical” solution by using mesh refinement, as the space of fields which belongs to H1(Ä)2 is not dense in the space of fields which belongs to H(curl, Ä)∩H(div, Ä)(this situation also occurs in 3D). This is major difficulty as far as the numerical computation of solutions is concerned.
Note that the numerical methods based on edge finite elements which are conforming in H(curl, Ä)[21], or those based on finite volumes on orthogonal meshes (in the 2D case) [18], allow one to approximate the solution. Indeed, the degrees of freedom of these methods are located on the edges of the mesh, and therefore they do not “carry” the geometrical singularities. Nevertheless, in order to get a precise knowledge of the solution close to the singularities, one must perform very significant mesh refinements (which makes them hard to use in some situations). What is more, in our case, using N´ed´elec’s second family of nodal finite elements [22], which are also conforming in H(curl, Ä), can be problematic.
As a matter of fact, the regularity required to define the associated moments (cf. [9]) is not automatically fulfilled by the solution of the time-dependent Maxwell equations.
Still, for most applications, computing the behavior of the solution outside of a neigh- borhood of the singularity is not sufficient. It is required that one computes the solution precisely as close to this singularity as is possible.
In order to achieve sufficient precision, one can a priori try to use singular functions methods. These methods, which have been developed for elliptic problems (see the earlier works in [12]), consist of augmenting the basis of numerical functions by adding a singular function (H1-conforming at the reentrant corner). They are widely used when the lack of regularity of the solution leads to a slow convergence of the nodal finite element method.
In this case, a carefully chosen refinement of the mesh would have been sufficient, and the singular function method can be viewed as an alternative. Here, the key point is that the numerical basis, with or without the addition of singular functions or mesh refinement, already spans the whole set of solutions when the mesh size goes to zero.
Unfortunately, this method cannot be applied to solving Maxwell’s equations in a singular domain, as the solution computed by a nodal finite element method does not span the whole space of the physical solutions (cf. the density problem mentioned above). Once again, it is not a matter of speed of convergence, and a mesh refinement or the addition of a singular function does not improve the overall numerical method.
To cope with this difficulty, we have developed a new method for solving time-dependent Maxwell equations in singular domains, called the Singular Complement Method (SCM).
One of its advantages is that it can easily be included in already existing numerical codes, which allows one to increase their domain of application. As a matter of fact, it makes them usable in the presence of geometrical singularities simply by adding a small number of functionalities, without having to rewrite them in their entirety.
The SCM is based on an orthogonal decomposition of the solution into a regular part and a singular part. The original ideas can be found in the works of Grisvard (see [14], [15]
and the references herein). Let us consider the Laplace equation with a right-hand side in L2(Ä). Its solution belongs to H2(Ä)when the domain is smooth (or convex), but it only
belongs to H1+s(Ä), with 1/2<s<1, when there are geometrical singularities (see [15]
for a 2D domain, and also [10], [14] for a 3D domain). Relying on an ad hoc decomposition of L2(Ä), Grisvard showed that it is possible to split the solution in a regular (in H2(Ä)) part and a singular part. Along the same lines, let us mention another approach developed by Hazard et al. (see [8], [17]) for the treatment of the time-harmonic Maxwell equations, which is also based on a decomposition of the space of solutions. Their decomposition is however slightly different from ours, thus leading to a different numerical scheme.
We introduced an orthogonal decomposition of the solution of Maxwell’s equations, which is mathematically analyzed in [5]. We recall the principles of the method in Section 2.
Hereafter, we build a constructive method for solving the original problem, and its associated numerical algorithm, which are detailed in Sections 3 and 4. The algorithm makes use of the explicit knowledge of the expression of the singularities near the reentrant corners. In Section 5, we present some numerical results which illustrate the efficiency of the SCM, applied to a number of realistic devices.
2. A MATHEMATICAL ANALYSIS OF THE PROBLEM
2.1. Maxwell’s Equations
Let us consider a bounded, connected, and simply connected open subsetÄofR2, the boundary of which is called0; let the infinite cylinderÄ×Rbe the physical domain. Let ν=(νx, νy,0)Tbe the outward unit normal to the domain, with the exception of the infinite edges. If we let c andε0be, respectively, the speed of light and the dielectric permittivity, Maxwell’s equations read
∂E
∂t −c2curlB= −1 ε0
J, (1)
∂B
∂t + curlE=0, (2)
divE= ρ ε0
, (3)
divB=0, (4)
whereE is the electric field,Bis the magnetic induction, andρandJ are the charge and current densities. These quantities depend on the space variable x and on the time variable t. Note that the above system of equations (1–4) is considered insideÄ×R.
These equations are supplied with appropriate bounday conditions. In order to simplify the presentation, let us assume first that the boundary is a perfect conductor. In Section 4, the extension to the Silver–M¨uller boundary condition will be handled: it can model either an absorbing medium outside of the domain or an incident wave. For the moment, let us take the conditions
E×ν=0 on0×R, (5) B·ν=0 on0×R. (6) The charge conservation equation is a consequence of equations (1–3) and reads
∂ρ
∂t +divJ =0. (7)
Last, initial conditions are provided (for instance at time t=0)
E(·,0)=E0, (8)
B(·,0)=B0, (9)
where the couple(E0,B0)depends only on the variable x.
In what follows, it is assumed that both the data and the initial conditions do not depend on the transverse variable z. Then the original problem can be identified with a problem the domain of which is a section of the infinite cylinder, that isÄ. As a matter of fact, the set of first order in time equations (1–4) can be rewritten equivalently as two decoupled sets of first order in time equations. If, for the fieldZ=(Zx,Zy,Zz)T ofR3one uses the notation Z=(Zx,Zy)T, the first set of equations is of unknowns(E,Bz), i.e., the so-called TE mode, with data J andρ, while the second set is of unknowns(Ez,B)with datum Jz
(the TM mode).
In the 2D case, let us note that there exists a scalar curl operator, denoted by curl, and a vector curl operator, denoted by curl. Both systems can be equivalently formulated as second order in time systems (see [6]). The TE mode can be written as
∂2E
∂t2 +c2curl curl E= −1 ε0
∂J
∂t, (10)
div E= ρ ε0
, (11)
∂2Bz
∂t2 −c21Bz = 1 ε0
curl J. (12)
Letτ=(νy,−νx)T be the tangential vector associated to the normal vectorν=(νx, νy)T. With these notations, the perfectly conducting boundary condition can be written, for the electric field
E·τ =0, (13)
and for the magnetic induction
∂Bz
∂ν − 1 c2ε0
J·τ =0. (14)
In addition to these conditions, the second order in time system of equations is closed with the help of initial conditions on∂E/∂t and∂Bz/∂t
∂E
∂t(·,0)=c2curl Bz0− 1 ε0
J(·,0), (15)
∂Bz
∂t (·,0)= −curl E0. (16)
The TM system could be written in the same way. In this paper, we focus on the above system, keeping in mind that one can also handle the other system with no additional difficulty (see [5] for more details). Also, we assume in the following that the domainÄ has a single reentrant corner. We refer the reader to [5] for the more general case, that is
a polygon with several reentrant corners. Again, there is no additional difficulty, with the exception of the formulas, which become a little bit more complicated.
The scalar product of L2(Ä)is denoted by(f |g)0=R
Ä f g dx and the related norm is k · k0. Let us introduce the functional spaces
L20(Ä)= {f ∈ L2(Ä), (f |1)0=0}, (17) H0(curl, Ä)= {u∈ H(curl, Ä),u·τ =0 on0}, (18) V = {u∈ H0(curl, Ä),div u=0}; (19) V is equipped with the canonical scalar product of H (curl,Ä):
(u,v)7→(u|v)0,curl
def=(u|v)0+(curl u|curl v)0.
In the present case, or more generally in a nonconvex domain with several reentrant corners, the space V is not included in H1(Ä)2any more (see [14] for instance). It is thus natural to introduce the regularized space VRof V :
VR = {u∈ H1(Ä)2,div u=0,u·τ =0 on0} =V ∩H1(Ä)2. (20) It is proved in [1] (see also the Introduction) that the Bzcomponent, as the solution of a wave equation, always belongs to H1(Ä), even in a nonconvex domain. It can therefore be computed numerically without any problem. On the contrary, the electric field E, which is the solution of Maxwell’s equation (10), (11), does not belong to H1(Ä)2as would be the case in a convex domain, but it only belongs to V . This is the reason why we consider only the computation of the field E in what follows.
Let us introduce the time-dependent problem:
Given a current J(t)such that div J=0, and two initial data E0 and E1, find E(t) ∈ H(curl, Ä)such that
∂2E
∂t2 +c2curl curl E= −1 ε0
∂J
∂t, (21)
div E=0, (22)
E·τ =0 on0, (23)
with the classical initial conditions E(0)=E0 and∂E/∂t(0)=E1. It can be written in a variational form:
Find E(t)∈H0(curl, Ä)such that d2
dt2(E|F)0+c2(curl E|curl F)0 = −1 ε0
µ∂J
∂t
¯¯¯¯F
¶
0
∀F∈H0(curl, Ä), (24)
div E=0, (25)
with the same initial conditions. One could also formulate the equivalent of (21)–(23) and of (24), (25) in H0(curl, Ä)∩H(div, Ä)or in V .
Thanks to [20], these exists one and only one solution to these problems.
Remark 2.1. The problem (21)–(23) is written in the absence of charges, that is with div E=0. One can consider the problem with charges, where (22) is replaced, for ρ ∈ L2(Ä), by div E=ρ/ε0, and where div J=0 is replaced by∂ρ/∂t+div J=0. As a matter of fact, it can still be reduced to the divergence-free problem. In order to do so, one can, for instance, let E?=E− ∇ψ, withψthe unique element of H01(Ä)that satisfies1ψ=ρ/ε0; this equation can be solved using its usual variational counterpart. This method is often called the Poisson correction (see, among others, [7] and [18] for numerical experiments).
In the present case,ρis a time-dependent function, and it is best to avoid the solution to a linear system at each time step. To address this difficulty, we have built a method which is a generalization of the method we apply in this paper, that is, the decomposition of the solution in a regular part and a singular part in V , to the case of fields with a nonvanishing divergence (see [2]).
2.2. A Decomposition of the Solution in Regular and Singular Parts
Let us briefly recall, without proof, some useful theoretical results in order to understand better the construction of the numerical method. The reader is referred to [5] for a thorough study.
The underlying principle of the method consists of relating the singular solutions of Maxwell’s equations to those of the Laplace problem, the properties of the latter having been investigated in a detailed manner (cf. [10, 14, 15]). On the one hand, there is the orthogonal decomposition of L20(Ä)
L20(Ä)=1(8R)⊕⊥ SN, (26) where1(8R)is the range of8Rby the operator1, with
8R =
½
φ∈ H2(Ä)/R,∂φ
∂ν
¯¯¯¯
0=0
¾ ,
and SNis characterized as the set of distributionsψ∈ L20(Ä)such that
1ψ =0 inD0(Ä), (27)
∂ψ
∂ν =0 on0. (28)
Thanks to Grisvard [15], SN is a finite dimensional vector space, it dimension being equal to the number of reentrant corners, that is 1 in our case. Let us call pSits basis.
On the other hand, we proved the following result:
LEMMA2.1. The scalar operator curl is an isomorphism from V onto L20(Ä). In addition, the L2-norm of the curl defines on V a norm, which is equivalent to the normk·k0,curl.
Let V be equipped with the scalar product induced by v7→kcurl vk0. In this case, the isomorphism of Lemma 2.1 preserves orthogonality. Having proved that the space VR, as well as its range by the scalar curl operator, are closed in V and L20(Ä), respectively, it is valid to introduce their orthogonal supplementary subspaces. This allows one to conclude that the orthogonal of curl VRin L20(Ä), denoted by(curl VR)⊥, is equal to SN. In this way, one obtains a result concerning the orthogonal decomposition of vector fields in V :
THEOREM2.1. The space V can be split in the orthogonal sum V=VR
⊕⊥ VS, where VSis a vector space of dimension 1, defined by curl VS=SN.
The electric field, solution of (21)–(23), depends continuously on the time variable with values in V [1]. Then, at any time t , one obtains the (orthogonal) decomposition
E(t)=ER(t)+ES(t). (29) The field ER∈VR is called the regular part of the solution, whereas the field ES∈VS is called the singular part. Moreover, the space VSis of dimension 1, so one may write
E(t)=ER(t)+κ(t)vS, (30) with vSa basis of VSandκa function which is smooth in time (at least continuous, cf. [1]).
Remark 2.2. 1. In the particular cases when ES=0 (i.e., the domain is convex), or when the singular coefficientκis zero, one has obviously E=ER ∈ H1(Ä)2. This property must be preserved numerically: this fact is illustrated in Section 5.
2. When the domainÄhas K reentrant corners, dim(VS)=K . Then, for(vSj)1≤j≤K a basis of VS
E(t)=ER(t)+ X
1≤j≤K
κj(t)vSj, (31) where(κj)1≤j≤K are K smooth functions (cf. [1]).
3. A NUMERICAL METHOD TO COMPUTE THE SOLUTION
Starting from the orthogonal decomposition obtained previously, it is possible to build a method, which allows one to compute numerically the solution.
Remark 3.1. Let us stress the fact that this method can be easily included into already existing codes, without the costly procedure of rewriting them entirely. Here, we are simply stating that a code which computes the solution of Maxwell’s equations in convex domains actually computes the regular part, as the singular part is always equal to zero in this case.
Thus, the method we present here provides an extension of the range of the code to the case of nonconvex (and nonsmooth) domains.
It can be summarized in two steps:
1. Determination of a basis of VS. One must solve a static problem. The computations are carried out only once as an initialization procedure.
2. Solution to the time-dependent problem (24), (25). It will then be enough to couple the classical method, which is already available, to the solution of an additional equation, of unknownκ(t).
Those steps are enumerated in the following subsections.
3.1. Determination of vS, a Basis of VS 3.1.1. Principle of the Method
To compute vS, the isomophism of Lemma 2.1 is used. The framework of the algorithm is as follows:
• First step
The space SN is of dimension 1: one computes first a basis of SN, that is an nonvanishing element pS of L20(Ä), which satisfies to
1pS =0 inÄ, (32)
∂pS
∂ν =0 on0. (33)
• Second step
One must then to look for vS ∈ H(curl, Ä), the solution of
curl vS = pSinÄ, (34)
div vS =0 inÄ, (35)
vS·τ =0 on0. (36)
Instead of using the direct solution to (34)–(36), it is more practical to make use of another isomorphism (see [5]), which is analogous to the one of Lemma 2.1. It shows that, to vS∈V , there corresponds one and only one potentialφS∈ H1(Ä)/Rsuch that
−1φS = pSinÄ, (37)
∂φS
∂ν =0 on0. (38)
Now, asφSis sufficiently smooth (i.e., with regularity H1), one can easily solve this problem with the help of a variational formulation. The computation of vS ∈VSthen stems from the identity vS=curlφS.
3.1.2. A Numerical Solution Obtained by Substructuring
Thanks to the above results relating the singular field vS to pS, it is possible to derive some useful information about the expression of the singularities in the neighborhood of the reentrant corner (recall that its counterpart is well known for pS). In order to benefit from this explicit knowledge, the computational algorithm is built using a substructuring approach.
Let us begin with the computation of pS. It can be viewed as a generalization of the method originally developed for the Laplace problem by Givoli and Keller [13, 19], their transmission operator being a particular instance of the capacitance operators (see for in- stance [11]). With this approach, one gets an explicit expression of pS in a neighborhood of the reentrant corner. Outside of this neighborhood, pS is smooth, and it can therefore be computed with the help of a classical variational formulation. It should be noted that the information corresponding to the “exact” knowledge of pSclose to the reentrant corner allows one to preserve (numerically) the orthogonality between the regular and singular parts of the solution. This is not the case anymore if one chooses to regularize pS“locally,”
by substracting to it its most singular part, that is, this most singular part multiplied by a smooth cut-off function (cf. [15]).
Let us take the domainÄsuch as it is pictured on Fig. 3. Its reentrant corner is locally made up of two segments, which intersect at the corner with an angleπ/α,1/2< α <1.
FIG. 3. Shape of the domainÄ.
A partition ofÄinÄcandÄeis introduced, whereÄcstands for an open angular sector of radius R centered at the reentrant corner, and whereÄe is the open domain such that Äc∩Äe= ∅and ¯Äc∪įe=į. Last, let0c(respectively,0e) denote the boundary ofÄc (respectively,Äe), which is split inB∪0˜c(respectively,B∪0˜e), withB=0c∩0e.
In this paper, either a (superscript)coreis added to refer to the restriction of quantities toÄcorÄe.
First Step: Computation of pS.
Let us consider the detailed computation of pS, which is a nonvanishing element of SN. It can be further divided into four substeps. The first two are “formal” (but still mathematically justified); the other two allow one to compute pSnumerically. Substeps 1 and 2, respectively, consist in finding an expression of pcSas a series, then in determining a formula which allows one to recover the coefficient of the series as a function of pcS|B, and finally in computing a transmission operator on the interfaceB. After these substeps have been completed, one can pose the problem inÄe, the solution of which is peS: it is then computed in Substep 3. From that point on, in Substep 4, one computes the coefficients of the series, using pcS|B=peS|B, to obtain pcS.
1. An expression of the restriction pcS:
Using the separation of variables, one can find analytically the solution in a neigh- borhood of the reentrant corner. One gets, using the polar coordinates (centered at the corner) r≤R,0≤θ≤πα,
pcS(r, θ)= X
n≥−1
Anrnαcos(nαθ), with A−16=0. (39)
Every Ancan be written as an integral of pcS|B. Let us recall here the expression of the coefficients An
A0= α π
Z π/α
0
pcS(R, θ)dθ, (40)
A1= 2α π R−α
Z π/α
0
pcS(R, θ)cos(αθ)dθ−R−2αA−1, (41)
An = 2α π R−nα
Z π/α
0
pcS(R, θ)cos(nαθ)dθ,∀n≥2, (42) where A1is given as a function of A−1, which, by definition, is not equal to zero.
2. The capacitance operator T :
Letνcdenote the unit outward normal toÄc. Thanks to (40)–(42), one can define the capacitance operator T : pcS|B7→ ∂∂νpcSc|B, by
T¡ pcS¢
= 2α2 πR
X
n≥1
n
½Z π/α
0
pcS(R, θ0)cos(nαθ0)dθ0
¾
cos(nαθ)−2α A−1
Rα+1cos(αθ).
(43) If we let T1stand for the first term of the right-hand side, then
T¡ pcS¢
=T1¡ pcS¢
−2α A−1
Rα+1cos(αθ). (44) 3. Computing the solution peSto the exterior problem:
With the help of the transmission conditions peS|
B=pcS|
B and∂∂νpeSc|B=∂∂νpcSc|B, one gets the boundary condition for the exterior problem onB. Letνedenote the unit outward normal toÄe, the exterior problem then reads
Find peS ∈ H1(Äe)/Rsuch that
1peS =0 inÄe, (45)
∂peS
∂νe =0 on ˜0e, (46)
∂peS
∂νe +T1
¡peS¢
=2α A−1
Rα+1cos(αθ)onB, (47) which can be written in a variational form
¡∇peS¯¯∇q¢
0,Äe+ Z
B
T1¡ peS¢
q dσ = 2αA−1
Rα+1 Z
B
cos(αθ)q dσ, ∀q ∈H1(Äe)/R.
(48) (· | ·)0,Äe stands for the scalar product of L2(Äe). Clearly, the bilinear form(p,q)7→
R
BT1(p)q dσis symmetric positive. Thus, for a given A−1, the above exterior problem is well-posed. In order to solve the exterior problem numerically, a triangular mesh of Äe is provided. The space H1(Äe)/Ris discretized with the help of Lagrange Finite Elements: let Vhe be the space of H1-conforming Finite Element functions thus generated, and let peh be the associated discrete solution. It can be written in the form peh=PNe
i=1piλi, where(λi)1≤i≤Ne are the basis functions of Vhe. After the discretization, the variational formulation (48) can be written as a linear system:
¡KÄe+KB¢
peh =F. (49)
Here, peh also stands for the vector ofRNe of entries pi. The matrixKBis “locally”
full but, being defined only on the interfaceB, its size remains small. Forλi andλj
the support of which intersects the interface, the entry(KB)i,jreads 2α2
πR X
n≥1
n
½Z π/α
0
λi(R, θ)cos(nαθ)dθ Z π/α
0
λj(R, θ)cos(nαθ)dθ
¾ ,
4. The computation of the coefficients Anis carried out with the help of (40)–(42). Thus, one can reconstruct the solution in the whole domain (but not at the reentrant corner, which belongs to the boundary).
Remark 3.2. If the Maxwell equations are coupled with a particle method, such as Vlasov’s equation, computing the trajectography requires that the electromagnetic field is known at every possible position of the particles, often very close to the boundary, but never on the boundary.
Second Step: Computation of vS
• Computation of the scalar potential: As mentioned earlier on, vSis computed via its scalar potentialφS. One solves first the system (37), (38) in a manner which is similar to the one kept for pS.
1. The local solutionφcSreads:
φcS = −X
n≥1
Bn
nαrnαcos(nαθ)− X
n≥−1
An
4nα+4rnα+2cos(nαθ), (50) where the coefficients(Bn)n≥1can be expressed as functions of the traceφcSonB, that is:
n=1 B1= −2α2 πRα
Z π/α
0
φcS(R, θ)cos(αθ)dθ
− µ α
4−4αA−1R2−2α+ α
4+4αA1R2
¶
, (51)
n≥2 Bn = −2nα2 πRnα
Z π/α
0
φcS(R, θ)cos(nαθ)dθ− nα
4nα+4AnR2. (52) 2. The capacitance operator t is
t¡ φcS
¢=T1¡ φcS
¢−1 2
Z R 0
pcS(r, θ)dr+ α
2−2αA−1R1−αcos(αθ). (53) The exterior problem, of solutionφeS, which can be written similarly to peS, is equivalent to the variational formulation:
FindφeS ∈H1(Äe)/Rsuch that
¡∇φeS¯¯∇ψ¢
0,Äe +R Z π/α
0
T1
¡φSe
¢ψ(R, θ)dθ
=¡ peS¯¯ψ¢
0,Äe +1 2R
Z π/α
0
½Z R 0
pcS(r, θ)dr
¾
ψ(R, θ)dθ
− α
2−2αA−1R2−α Z π/α
0
cos(αθ)ψ(R, θ)dθ, ∀ψ∈ H1(Äe)/R. (54) The solution to the exterior problem can be obtained by the Finite Element dis- cretization of the formulation (54). Let φhe denote the discrete solution in Vhe, that isφhe=PNe
i=1φiλi. It is the solution to the linear system
¡KÄeφhe+KB¢
φeh=MÄepeS+G, (55)
whereMÄe stands for the mass matrix, that is, the matrix associated to the L2scalar product inÄe. The matrix being identical to that of (49), this formulation with the potential requires only minor modifications. The coefficients Bn are then computed with the help of (51), (52), and the potential is thus known everywhere except at the reentrant corner.
• Computation of vS:
One simply gets vS by taking the vector curl ofφS. 1. The local solution vcS:
vcS =X
n≥1
Bnrnα−1
Ãsin(nαθ) cos(nαθ)
! +X
n≥−1
Anrnα+1 Ã nα
4nα+4sin(nαθ)
nα+2
4nα+4cos(nαθ)
!
, with B16=0. (56) 2. The exterior solution veS:
One solves the equation of unknown veS which reads:
Find veS∈(H1(Äe)/R)2such that
¡veS¯¯λ¢
0,Äe =¡
curlφeS¯¯λ¢
0,Äe,∀λ∈(H1(Äe)/R)2. (57) Numerically, one does not simply perform vS=curlφS; some straightforward interpola- tion is added, so that an approximation vhSis at hand. For that, the whole domain is meshed (in our case, there remains only to meshÄc, in an admissible manner onB).
Thus, in order to compute a numerical approximation of vS, the basis of VS, (56) is inter- polated at the vertices insideÄcto get vch (the coefficients An and Bn have already been computed). To compute veh, (57) is discretized
MÄeveh =RÄeφhe, (58) whereRÄestands for the curl matrix, associated to the term(curlφeS |λ)0,Äe.
Remark 3.3. The analytical solutions pcS,φcS, and vcSare known via series (which con- verge, the solutions being in L2(Äc)), they must be truncated for the numerical computa- tions. However, in practice, the series converge very rapidly (spectral convergence), which means that a small number of terms (less than 10) are sufficient to compute a solution accurately.
Remark 3.4. If, given f ∈L20(Ä), one wants to solve the (stationary) curl–div problem:
Find u∈ H(curl, Ä)such that
curl E= f, (59)
div E=0, (60)
E·τ =0 on0, (61)
one simply must split the solution of (59)–(61) as E=ER+κvS, with ER∈VR,vS the (known) basis of VSandκa constant. The orthogonality of ERand vS yields
(curl E|curl vS)0=(f |curl vS)0=κ(curl vS|curl vS)0.
Using (34), one finally obtains
κ = (f | pS)0
kpSk20
.
The computation of the regular part ERcan then be carried out with the help of a classical numerical method. Thus, it is clear that only a few modifications are required, as long as a code, which computes the solution to similar problems in convex domains, is already at hand.
3.2. Solution to the Time-Dependent Problem 3.2.1. The Variational Formulation
A new variational formulation of the problem (24), (25) is introduced, using the orthog- onal decomposition of V=VR ⊕⊥ VS, and of the solution E(t)=ER(t)+κ(t)vS(t) (cf. (30)). In an equivalent manner, we add to the space of test functions VR the func- tion vS. With this orthogonal decomposition, the classical formulation (24) can be written Find ER ∈VRsuch that
d2
dt2(ER|FR)0+c2(curl ER |curl FR)0
= −1 ε0
d
dt(J|FR)0−κ00(t)(vS |FR)0,∀FR ∈VR. (62) This formulation projects the solution of Maxwell’s equations on the space VR of the regular fields and carries the singularity onto the space VS. A by-product is the additional unknownκ00(t), the second derivative ofκwith respect to the time variable. Therefore, one must add an extra equation, obtained for instance by taking vSas a test function.
d2
dt2(ER|vS)0+κ00(t)kvSk20+c2κ(t)kpSk20 = −1 ε0
d
dt(J|vS)0. (63) Remark 3.5. In the more general case of a domain with K reentrant corners, the solution has the form (31). The space of test functions VR is then completed with the K functions (viS)1≤i≤K. The formulation (24) then reads
Find ER ∈VRsuch that d2
dt2(ER |FR)0+c2(curl ER |curl FR)0
= −1 ε0
d
dt(J|FR)0− X
1≤j≤K
κ00j(t)¡ vSj¯¯FR
¢
0, ∀FR ∈VR. (64)
The K additional unknowns (κ00j(t))1≤j≤K appear. The above system is completed in this case with K additional equations, where the K test functions are(viS)1≤i≤K. Thanks to the orthogonality of regular and singular fields, one gets
d2 dt2
¡ER¯¯viS¢
0+ X
1≤j≤K
κ00j(t)¡ vSj¯¯viS¢
0+c2κi(t)¡ pSj¯¯piS¢
0
= −1 ε0
d dt
¡J¯¯viS¢
0,1≤i ≤K. (65)
3.2.2. Discretization of the Formulation (62), (63)
One proceeds in two steps: first a semidiscretization in space, then a discretization in time.
Let us consider the semidiscretization in space of the variational problem (62), (63).
The triangle mesh is the (admissible) union of the meshes overÄeandÄc. Let VRh⊂VR
be the space of discretized test functions, and let Eh(t)=EhR(t)+κ(t)vS be the discrete solution. One knows that EhR(t)=P
iEiRλi, whereλiare the basis functions of the Finite Element Method. Here,λi denotes a vector basis function: one of its component is equal toλi, and the other is equal to zero. If it is additionaly assumed that vS is known exactly, the orthogonality relationships still hold and the semidiscretized variational formulation is written (with the addition of the indexh) in the same way as (62), (63).
Remark 3.6. 1. The semidiscrete system derived from (62), (63) is coupled via the termsκ00(t)and EhR(t). One can obtain a decoupled version with the help of the (orthogonal) projection Phon VRh with respect to the scalar product of L2(Ä)2, by taking vS−PhvSas a test function instead of vS. Equation (63) is then replaced by
κ00(t)kvS−PhvSk20+c2κ(t)kpSk20
=c2¡
curl EhR¯¯curl(PhvS)¢
0− 1 ε0
d
dt(Jh¯¯vS−PhvS)0. (66) Actually, if one considers the linear system resulting from (62), (63), this can be viewed as the usual factorization which is performed to provide a block triangular matrix. The functionκ is then computed by solving the ordinary differential equation (66). According to the regularity of vS[5], the estimation of the coefficient in front ofκ00(t)yields
kvS−PhvSk20≤Cεh2α−2ε, ∀ε >0,whereπ
αis the angle at the reentrant corner. (67) This allows one to conclude that this differential equation is not stiff, and so that it can be solved with a classical discretization in time. Nevertheless, the numerical experiments that we have carried out with this approach (cf. [4]) yielded results which are less precise than the one obtained using (62), (63).
2. In order to simplify the presentation, we present the case of an internal approxima- tion, that is VRh⊂VR. This assumption is, however, not required. As a matter of fact, the formulations which we implemented in our codes are not internal. Indeed, we kept mixed formulations in which the divergence constraint is dualized (see [6]). Under these condi- tions, the discrete functions of the approximation space, called ZhR, are not divergence-free.
One must add a Lagrange multiplier ph∈Qh to dualize the discrete divergence condition div EhR=0, where the space Qhis chosen in such a way that the discrete inf–sup condition is satisfied. This mixed formulation consists of adding a(ph|div FR)0term to the formu- lation (62). The loss of the orthogonality between VSand ZhR (as ZhR6⊂VR) yields another term in (62), which reads
¡curl vS¯¯curl FhR¢
06=0, ∀FhR ∈ ZhR. (68)
