Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 7965642, 17 pages
http://dx.doi.org/10.1155/2016/7965642

Research Article

On the Coefficients of the Singularities of the Solution of Maxwell’s Equations near Polyhedral Edges

Boniface Nkemzi

Department of Mathematics, University of Buea, Buea, Cameroon

Received 26 August 2015; Accepted 17 December 2015

Academic Editor: Haipeng Peng

Copyright © 2016 Boniface Nkemzi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The solution fields of Maxwell’s equations are known to exhibit singularities near corners, crack tips, edges, and so forth of the physical domain. The structures of the singular fields are well known up to some undetermined coefficients. In two-dimensional domains with corners and cracks, the unknown coefficients are real constants. However, in three-dimensional domains the unknown coefficients are functions defined along the corresponding edges. This paper proposes explicit formulas for the computation of these coefficients in the case of two-dimensional domains with corners and three-dimensional domains with straight edges. The coefficients of the singular fields along straight edges of three-dimensional domains are represented in terms of Fourier series. The formulas presented are aimed at the numerical approximation of the coefficients of the singular fields. They can also be used for the construction of adaptive -nodal finite-element procedures for the efficient numerical treatment of Maxwell’s equations in nonsmooth domains.

1. Introduction

Unlike the regularity analysis for elliptic boundary value problems in domains with geometric singularities, where there exists a unified theory based on the shift theorem (see [1–3]), the regularity analysis of the solution of Maxwell’s equations has several interpretations. Most papers are concerned with the -regularity for nonconvex domains, and it is shown that the main singularity is the gradient of singular functions associated with the Laplace equation; see, for example, Birman and Solomyak [4], Bonnet-Ben Dhia et al. [5], Moussaoui [6], Hazard and Lohrengel [7], Hazard and Lenoir [8], and Lohrengel [9]. Costabel in [10] went further to address the -regularity for Lipschitz domains.

The -regularity of the solution of Maxwell’s equations has been considered, for example, by Nkemzi [11–14]. To be more precise, the asymptotic behaviour of the solution near axisymmetric edges and their efficient numerical treatment by means of the Fourier-finite-element method on graded meshes were analyzed in [11, 14]. In [12] the problem was considered in axisymmetric domains with conical points and the asymptotic behaviour of the solution near the conical points was analyzed. Here explicit representation formulas for the coefficients of the singularities were derived. In [13] Maxwell’s equations in polygonal domains were considered and formulas for the coefficients of the singularities were derived. The present paper considers Maxwell’s equations in three-dimensional domains with polyhedral edges and the main focus is on the explicit description of the coefficients of the singularities. Unlike in two-dimensional case and the case of conical points where the space of the singular solutions is finite dimensional and the coefficients of the singularities are some real numbers, in three-dimensional domains with edges the space of the singular solutions is infinite dimensional and the coefficients of the singularities are functions defined along the edges. Here the space of the singular solutions along polyhedral edges is completely described.

It should be noted that a more rigorous regularity analysis based on the shift theorem for the Maxwell equations in plane domains with corners and in polyhedral domains has been carried out by Costabel et al. in [15, 16]. They showed, using the classical Mellin analysis, a technique due to Kondratev [17], that, for a given current density in , , the electromagnetic fields either belong to the space or else can be split into a regular part in and an explicitly defined singular part up to some unknown coefficients. They represented the singular fields along the edge in a tensor product form. However, it has been shown (see, e.g., Heinrich [18]) that the numerical treatment of boundary value problems in three-dimensional domains with edges is more efficient if the singular solutions along the edges are expressed in nontensor product forms.

Asymptotic analysis of solutions of boundary value problems near geometric singularities is usually carried out by considering only the principal part of the differential operator with frozen coefficients; see, for example, [16, 17]. This is due to the fact that the qualitative behaviour of the solutions near geometric singularities depends only on the principal part of the operator, the geometry of the domain near the singularity, and the nature of the boundary conditions. On the other hand, it is known that the coefficients in the asymptotic expansion are linear continuous functions of the right hand side datum of the differential equation; see [2, 3]. Thus, formulas for the coefficients are derived by taking into account the exact differential operator under consideration. These coefficients, especially the leading coefficients, determine the actual strength of the singularities and are the principal indicators for material damage analysis, for example, fracture analysis. Hence, their computation is not only important for the numerical treatment of the boundary value problem but also important for practical applications.

In this paper we extend the -regularity analysis of the solution of time-harmonic Maxwell equations in plane domains with corners (see [13]) to three-dimensional domains with polyhedral edges. The main concern is to derive explicit computable formulas for the coefficients of the singularities. The singular functions along polyhedral edges are expressed in Fourier series and in nontensor product forms which are more suitable for constructing accurate finite-element solutions. The results presented here can be used to construct postprocessing procedures for the -nodal finite-element treatment of Maxwell’s equations in domains with geometric singularities and with optimal accuracy.

This paper is organised as follows. In Section 2 we introduce the boundary value problem and related function spaces and address the issue of existence and uniqueness of the weak solution. In Section 3 we formulate in Theorems 5 and 6 the main results concerning the regularity properties of Maxwell’s equations in two-dimensional domains with corners and three-dimensional domains with polyhedral edges, respectively. Sections 4 and 5 contain detailed proofs of Theorems 5 and 6, respectively.

2. The Boundary Value Problem and Functional Tools

We consider as model problem the electromagnetic fields ( the electric field and the magnetic field) of time-harmonic Maxwell’s equations in a simply connected and bounded domain with Lipschitz boundary containing an isotropic and homogeneous medium subject to perfect conductor boundary conditions [5, 19, 20]:

where the domain related parameters , , and are, respectively, the dielectric permittivity, magnetic permeability, and electric conductivity, is a given div ergence-free current density, that is, , is the pulsation of the electromagnetic fields, is the imaginary unity, and denotes the unit outward normal on the boundary .

If we suppose temporarily that the vector function is sufficiently smooth, then system (1) can be written as two decoupled systems in terms of the magnetic field and the electric field as follows:

where , , and .

It follows directly from (1), (2), and (3) that the solution of problem (2) can be derived from the solution of problem (3) and vice versa. Thus it suffices to solve or analyze only one of the problems. Subsequently we will dedicate our analysis to the boundary value problem (3). For the Hilbert space formulation of (3) we introduce the function spaces; see [19, 21]:

equipped with the norm

The variational formulation of problem (3) is as follows.

Find such that

where denotes the usual scalar product in the Hilbert space of complex-valued functions. The well-posedness of problem (6) is addressed by the following theorem; see also [19, 22, 23].

Theorem 1. Let be a bounded domain with at least Lipschitz continuous boundary . Suppose and is not an eigenvalue of the operator with electric boundary condition. Then, for any , there exists a unique solution of the variational problem (6). Moreover, the solution satisfies the estimate

We will always assume where necessary, without explicitly stating so, that the conditions of Theorem 1 are satisfied.

We observe that the operator from (2) and (3) is not elliptic. However, a widely used alternative formulation of the boundary value problem (3) is the so-called regularized formulation of the Maxwell equations; see [5, 7–9]. In fact, it is easily seen that the boundary value problem (3) is equivalent to the boundary value problem

in the sense that the solution of (3) solves (8) and vice versa. We notice that the operator is elliptic. The associated Hilbert space formulation for the boundary value problem (8) is as follows.

Find such that

where

The following theorem addresses the question of well-posedness of problem (9); see also [5, 9].

Theorem 2. Let be a bounded domain with at least Lipschitz continuous boundary . Suppose and is not an eigenvalue of the Dirichlet-Laplace operator on . Then, for any , there exists a unique solution of the variational problem (9). Moreover, the solution satisfies the estimate

The rest of this paper is dedicated to a rigorous regularity analysis of the solution of time-harmonic Maxwell equations in simply connected and bounded domains , , with Lipschitz boundary containing an isotropic and homogeneous medium subject to perfect conductor boundary conditions. We will systemically use the regularized formulation (8) and all derivatives should always be understood in the sense of distributions. First we state here one regularity result that is frequently quoted in the literature; see [21]. We will use the notation

Theorem 3. Let be a bounded domain with boundary . If is of class or if is a convex polygon in or a convex polyhedron in , then the relation

holds and on these spaces the norms and are equivalent.

On the other hand, if is a nonconvex polygon or polyhedron, then the space is a proper closed subset of the space and the following result holds; see [4, 6].

Theorem 4. The space can be split as a direct sum of the form

where denotes the space of functions spanned by the singular functions associated with the Dirichlet boundary value problem for the Laplace equation in .

An immediate consequence of the regularity Theorems 3 and 4 is that, in nonconvex polygons or polyhedrons, the weak solution of the boundary value problem (8) does not belong to the space and can therefore not be approximated by means of the usual -conforming nodal finite-element method.

3. Corner and Edge Singularities for Maxwell’s Equations

In this section we formulate the main results of this paper. The proofs which are very lengthy in nature will be carried out in subsequent sections.

3.1. Corner Singularities for Maxwell’s Equations

Here we consider the electric field of time-harmonic Maxwell’s equations in a simply connected and bounded domain with Lipschitz boundary , formally the variational solution of the boundary value problem

where and the parameter are given.

Now, suppose that the boundary of consists of finitely many disjoint analytic arcs , , such that , where the segments are numbered according to the positive orientation, that is, in anticlockwise direction. Let the endpoints of each be denoted by and let the solid angle at be denoted by , where . We denote by and (resp., and ) local polar (resp., Cartesian) coordinates attached to the vertex , such that

that is, is supported by the line and is on the line . Suppose that the domain coincides near each singular point with a circular sector with radius and angle ; that is,

The boundary will be represented subsequently as , where

We define with respect to the vertex a smooth truncation function which depends only on the distance from by

where is taken from (17); that is, . Furthermore, we define on each sector neighbourhood of the vertex the functions

where , and the parameter are taken from (15), is as defined in (19), and

Our main result on corner singularity is the following.

Theorem 5. For each and , let be the variational solution of the boundary value problem (15). Let , , , . If , , , then there exist coefficients such that the solution can be split as a sum in the form

with . The coefficients of the asymptotic expansion (22) are given explicitly by the formula

where the function is defined in (20). The constants and and the local Cartesian and polar coordinates , and , are as specified in (17) and (16). Moreover, there exists a constant independent of and such that

The proof of Theorem 5 will be carried out in Section 4; see Theorem 9.

3.2. Edge Singularities for Maxwell’s Equations

Let be a simply connected and bounded domain with Lipschitz boundary . For and , we consider the variational solution of the boundary value problem

Since we are interested only in the asymptotic behaviour of the solution near straight edges of the domain , we can assume, without loss of generality, that the domain is a prismatic cylinder; that is, it has the form with a real constant and a bounded domain with piecewise smooth boundary and such that for each , , and . In this way we can use the same notations as in Section 3.1 for . In particular, the edges of the domain are and the measure of the interior angle along the edge is , . We associate with each edge a wedge , where is as defined in (17). Further we introduce on the functions

where the functions , , and the parameter are taken from (25), is from (19), and , are as defined in (21). Obviously .

Our main result on edge singularity is the following.

Theorem 6. For and , let be the variational solution of the boundary value problem (25). Let , , , . If , , , then there exist unique functions and such that the solution can be split into a regular and a singular part in the form

The functions and are fixed kernels defined by

The coefficients and of the asymptotic expansion (27) can be expressed in Fourier series in the form

where the Fourier coefficients and are given explicitly by the formulas

Here, the function is as defined in (26). The constants and and the local Cartesian and polar coordinates , , , and are as defined in (17) and (16). In (27) the symbol “” denotes convolution product in the variable ; that is,

The proof of Theorem 6 will be carried out in Section 5; see Theorem 24.

4. Maxwell’s Equations in Two-Dimensional Domains with Corners

In this section, we consider in greater detail the structure of the solution of the Maxwell equations (15) in two-dimensional domains with corners and show how the results of Theorem 5 are derived.

4.1. Maxwell’s Equations in a Bounded Sector

For purely mathematical reasons we consider first a slightly modified boundary value problem for the Maxwell equations in a circular sector; see Figure 1. The results of this subsection are largely found in Nkemzi [13] and will be kept very brief.

Figure 1: A circular sector.

Let denote a circular sector in with radius , interior angle , and boundary ; see Figure 1. We assume that the Cartesian coordinate system of and is positioned such that the vertex of coincides with the origin and the side is supported by the -axis.

For , we consider the unique weak solution of the boundary value problem

Local polar coordinates and in with respect to the vertex are related to the Cartesian coordinates and ; namely,

Accordingly, the sector domain is transformed by the one-to-one mapping into the rectangle

in the polar coordinate system. By the transformation (33), each function defined on is mapped uniquely to some function defined on by

Similarly, each vector field defined on is mapped uniquely to some vector field defined on by

The boundary value problem (32) can be solved explicitly. In fact, we have the following result which can be verified by direct substitution.

Lemma 7. The weak solution of the boundary value problem (32) can be represented in Fourier series in the form

where the Fourier coefficients satisfy the relations

Using the explicit representation formulas (37)-(38) for the solution of the boundary value problem (32) and taking into account relation (36) one can derive various regularity properties for the solution.

The main result of this subsection is the following; see, for example, [13, 15, 16], for the proof.

Theorem 8. Let be a circular sector with angle , . Let , . Then for each the solution of the boundary value problem (32) has the following additional regularity properties:

(a)There exists a constant independent of and such that the

 That is, the condition is always satisfied.(b)If , (i.e., is convex), then and there exists a constant independent of and such that

(c)If , (i.e., ), then and there exists a constant independent of and such that

(d)If , , then can be split into a regular and a singular part in the form

 The coefficients are given explicitly by the formula

 Moreover, there exists a constant independent of such that

4.2. Maxwell’s Equations in Plane Domains with Corners

We can now make definite statements on the regularity properties of the solution of the Maxwell boundary value problem (15) in bounded domains with piecewise smooth boundary . We will use the same notations as in Section 3.1.

Let be the solution of (15). Then the function , where is the smooth truncation function from (19), belongs to the space and is the unique weak solution of the boundary value problem

where the function is as defined in (20).

We observe that problems (45) and (32) are similar and therefore their solutions have the same regularity properties as described in Theorem 8. On the other hand, the solution of problem (45) coincides near the vertex of to the solution of problem (15). Thus the two solutions have the same asymptotic behaviour near the vertex . Taking into consideration the fact that singularity is a local property and the technique for coupling local and global regularity properties (see [2, 18]), we obtain directly from Theorem 8 the following properties for the solution of problem (15).

Theorem 9. For each , let be the solution of the boundary value problem (15). Let , , , and . Then the solution has the following additional regularity properties:

(a) and there exists a constant independent of and such that

(b)If , , (i.e., is convex), then and there exists a constant independent of and such that

(c)If , , (i.e., for all ), then and there exists a constant independent of and such that

(d)If , , , then the solution can be split into a regular and a singular part in the form

 The coefficients are given explicitly by the formula

 where is defined in (20) and the local Cartesian coordinates and are defined in (16). Moreover, there exists a constant independent of such that

4.3. Some Auxiliary Boundary Value Problems with a Parameter

Let and be given functions. We consider in this subsection the following boundary value problems with parameter on the sector domain (see Figure 1), formulated in polar coordinates:

where the vector field from (52) in polar coordinates is linked to the vector field in Cartesian coordinates according to relation (36) and the scalar function from (53) is linked to the scalar function according to relation (35). The symbol denotes a positive real parameter. The interest on the boundary value problems (52) and (53) is purely mathematical and there is no evidence that these problems have any practical applications. The results obtained in this subsection will be used for the analysis of the regularity properties of the solution of Maxwell’s equations near polyhedral edges.

The main results concerning the regularity properties of the solution of the boundary value problems (52) and (53) are formulated in Theorems 10 and 11.

Theorem 10. Let , , and , , with . Then the boundary value problem (53) has a unique variational solution :

(a)If , then the solution and there exists a constant independent of , , and such that

(b)If and , then there exists a coefficient such that the solution can be split as a sum of a regular and a singular part in the form

 The coefficient is given explicitly by the formula

 Furthermore, there exists a constant independent of and such that

(i) as ,(ii) as .

For the proof of Theorem 10, see [24, pp. 171–174].

Theorem 11. Let , , and , , with . Then there exists a unique variational solution of the boundary value problem (52). If , , then there exist coefficients such that the solution can be split as a sum of a regular and a singular part in the form

The coefficients are given explicitly by the formula

Furthermore, there exists a constant independent of and such that

(i) as ,(ii) as .

The proof of Theorem 11 is very lengthy and will be broken down into several lemmas as follows.

Lemma 12. The solution of problem (52) can be represented in a Fourier series in the form

where the Fourier coefficients , , are given explicitly by the formulas

Lemma 13. Let

If , then there exists a constant independent of and such that

Proof. Application of Cauchy-Schwartz inequality and the substitution lead to the estimates

Lemma 14. Let

If , then there exists a constant independent of such that

(i)

(ii)

(iii)

Proof. Using the substitution one easily verifies the estimates

The following lemmas can be proved by analogy. We omit the proofs for the sake of brevity.

Lemma 15. Let

Then there exists a constant independent of and such that

Lemma 16. Let

Then there exists a constant independent of such that

(i)

(ii)

(iii)

Lemma 17. Let

Then there exists a constant independent of and such that

(i)

(ii)

(iii)

Lemma 18. Let

Then there exists a constant independent of and such that

(i)

(ii)

(iii)

The proof of Theorem 11 can now be summarised as follows.

Proof (Theorem 11). The results of Theorem 11 follow by combining Lemmas 12–18, taking note of the definition of the representation of norms of functions as series of norms of their Fourier coefficients by means of generalized Parseval identities; see, for example, [13, 18].

5. Maxwell’s Equations in Domains with Polyhedral Edges

In this section, we consider and analyze the Maxwell equations (25) in three-dimensional domains with polyhedral edges and prove Theorem 6.

5.1. Maxwell’s Equations in a Three-Dimensional Wedge

We consider first a three-dimensional domain of the form , where is a real constant and ; see Figure 1. We will use the same notations as in Section 4.1 for . Thus the boundary of can be represented in the form , where , , and , ; see Figure 2.

Figure 2: A three-dimensional wedge.

For a given vector field , let be the variational solution of the boundary value problem

We observe that the systems of trigonometric functions and () are orthogonal and complete in ; see [25, 26]. Thus functions can be characterized by their Fourier coefficients as follows.

Lemma 19. (1) Let . Then there exist in Fourier coefficients of defined by

and satisfying the relations

Moreover, Parseval’s identity holds in the form

(2) For , relations (84), (85), and (86) hold and additionally

For the study of the Fourier coefficients of the solution of the boundary value problem (83), we introduce on the spaces

where denotes the unit outward normal on the boundary . These spaces are equipped with the norm

Clearly, the spaces and endowed with the norm (89) are Hilbert spaces. Indeed, we observe the identities

Lemma 20. Let denote the Fourier coefficients of a function defined according to relation (84). Then for any .

Proof. It follows from (84) that . The completeness relationship (87) infers then that and consequently . Further, with the help of the triangle inequality and relation (87) we get the estimates

Hence, taking into account the definition of and the completeness relationship (87) we get , . The boundary conditions follow from the boundary conditions in .

Lemma 21. For , let be the solution of the boundary value problem (83). Let and denote the Fourier coefficients of and , respectively, defined according to (84). Then the functions , , satisfy the relations and and are the solutions of the boundary value problems

Proof. Problems (92) are obtained directly from (83) by substituting the functions and by their respective Fourier series defined according to (85), differentiating term by term and comparing coefficients. The assertions and follow from Lemma 20.

We observe that in local polar coordinates and problems (92) take the form

We will need the following notations:

where and , , are taken from (93). Obviously .

Theorem 22. Let be a circular sector with angle , , and let , . For each , let , , be the solutions of the boundary value problems (92). If , , then there exist coefficients and such that the solutions , , can be represented in the form

The coefficients and of the expansion (95) are given explicitly by the formulas

where the functions , , are as defined in (94). Moreover, there exists a constant independent of such that

Proof. Relations (95), (96), (97a), (97b), (97c), and (97d) are obtained by a straightforward application of Theorems 10 and 11 to the boundary value problems (93) with and taking note of the modified right hand side defined in (94).

Theorem 23. Let be a three-dimensional wedge and , , . For each , let be the solution of the boundary value problem (83). If , , then there exist unique functions and such that the solution can be split in the form

where

In (98), the symbol “” denotes the convolution product in the variable ; that is,

The coefficients and are as defined in Theorem 22.

Proof. The expressions (98), (99), and (100) are direct consequences of Lemma 21 and Theorem 22, taking into consideration relations (35) and (36). The inequalities (97c) and (97d) and Parseval’s identity (86) imply that the Fourier coefficients , , satisfy the estimate

Hence, ; see [24, Theorem ]. The inequalities (97a) and (97b) lead to the estimates

which by the generalized Riesz-Fischer theorem imply the existence of functions and whose Fourier coefficients are and , respectively.

5.2. Singularities near Polyhedral Edges

In this subsection, we consider and analyze the regularity properties of the solution of the Maxwell equations (25) in general three-dimensional domains with straight edges bounded by plain faces, that is, polyhedral edges. In fact it would be sufficient for us to consider three-dimensional domains of the form , where is a real constant and is a general bounded domain with piecewise smooth boundary . We will use the same notations as in Section 3 for and .

Thus the edges of are , , and the boundary is the union of the disjoint faces , , and the two bases and . We assume that, for , and . We associate with each edge a three-dimensional wedge , where is a circular sector; see (17) and (18). Thus the boundary of is the union of the disjoint faces , , , , and . Furthermore, we define on each a smooth truncation function ; see (19).

For and , let be the variational solution of the Maxwell equations (25). Then the function which is defined on the wedge belongs to the space and is the unique weak solution of the boundary value problem

where the function is given by

We observe that problems (83) and (103) are similar and therefore their regularity properties are described by Theorem 23. On the other hand, since the solution of problem (25) coincides with the solution of problem (103) near the edge of , Theorem 23 also addresses the regularity properties of the solution of (25). Thus we obtain the following theorem.

Theorem 24. For and , let be the variational solution of the boundary value problem (25). Let , , and , . If , , , then there exist functions and such that the solution can be represented in the form

The functions and are fixed kernels given by

The coefficients and are defined explicitly by

where the coefficients and are as defined in (30).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Alexander von Humboldt Foundation (AvH), Bonn, Germany, and the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy.

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