A coercive mixed formulation for the generalized Maxwell problem

https://doi.org/10.1016/j.cam.2021.113787Get rights and content

Abstract

A coercive mixed variational formulation on H0(curl;Ω)×H(div;Ω) is proposed for the generalized Maxwell problem which typically arises from computational electromagnetism. The mixed variables are the electric field and a pseudo electric displacement field. The well-posedness of the mixed variational problem is proven in the general settings (multiply connected domain of Lipschitz-continuous boundary with a number of connected components, filling with discontinuous, anisotropic and inhomogeneous media); more importantly, the coercivity is established. A conforming finite element discretization is further proposed, where the electric field is approximated by H(curl;Ω)-conforming edge element while the pseudo electric displacement field by H(div;Ω)-conforming flux element. Error estimates are obtained, and in particular, the method produces an L2 curl-convergent approximation and more importantly, an L2 div-convergent approximation for the solution.

Introduction

Given a three-dimensional domain ΩR3. Two classical stationary models of the Maxwell equations are formulated as the curl-div first-order system and the curlcurl-div second-order system [1], [2], [3]:

The curl-div system: curlu=f1,div(ɛu)=g1in Ω.

The curlcurl-div system: curl(μ1curlu)=f2,div(ɛu)=g2in Ω.The unknown variable u represents the electric field and the tensor μ,ɛ reflect anisotropic inhomogeneous properties of material filling Ω, i.e., magnetic permeability and electric permittivity. Both systems can be to some extent transformed into each other mutually and can also be written as a general formulation, called curlcurl-grad div system (called the generalized Maxwell problem in this paper), as follows: curl(μ1curlu)ɛdiv(ɛu)=f3in Ω.A third example of the above curlcurl-grad div system may come from fluid–structure interaction, e.g., divu=χ,curlu=f4in Ω.These systems have played a central role in theoretical and numerical issues as well as practical applications (e.g., electrical engineering applications1) about computational electromagnetism. These systems also constitute computational models of time-dependent problems and eigenproblems, and we shall deal with the following generalized time-harmonic Maxwell problem, which covers the usual time-harmonic Maxwell equations and the discretization of the time-dependent Maxwell equations, for a given real number γR far from the eigenvalues of the partial differential operator of (1.3), curl(μ1curlu)ɛdiv(ɛu)+γɛu=f5in Ω,and shall also deal with the usual Maxwell eigenproblem curl(μ1curlu)=λɛuin Ω.

The purpose of this paper is to develop a mixed formulation and its finite element method for (1.3). How to design the finite element method for (1.3) is very crucial and important, because all these systems (1.1), (1.2), (1.3), (1.4), (1.5), and (1.6) can be unified into one approach for numerical solutions. Usually, either of the following three variational formulations can act as such unifying approach. The first unifying approach is the curl-div displacement-based variational formulation [5], see also Section 3.2, where the main part in the bilinear form reads Ωμ1curlucurlv+Ωdiv(ɛu)div(ɛv).The second unifying approach is the mixed formulation [6], [7], [8], [9], where the main part in the bilinear form involves Ωμ1curlucurlv+Ωɛvp,Ωɛuq,where the scalar variable p, known as the multiplier, is related to the div operator. The third unifying approach is the perturbation formulation [10], where the main part in the bilinear form is Ωμ1curlucurlv+δ0Ωɛuv,where the parameter δ0>0 and δ00.

The displacement-based approach (1.7) is natural, as far as these systems (1.1), (1.2), (1.3), and (1.4) are concerned. It solves the solution in the space H0(curl;Ω)H(div;ɛ;Ω). It results in a symmetric, positive definite algebraic system. It has been extensively employed in theoretical analysis. It could produce L2 curl-convergence and L2 div-convergence in the finite element solution. However, the challenge for (1.7) is that whenever the exact solution is singular not belonging to H1(Ω) space, the finite element solution may not correctly converge, in general, whatever Lagrange elements or discontinuous elements are used. In this paper, the singular solution is understood in the following sense: u,curlu(Hs(Ωl))30s1,Ω=l=1LΩl.The regularity of the solution depends on those of both the domain and the right-hand sides such as f and g. In practice, the domain is rather complicated, and the reentrant corners and edges give rise to singular solutions, even if the right hand sides are very smooth. For general inhomogeneous media, the solution may be more singular, i.e. it is only piecewise Hs in the sense of (1.10). For such challenge, there have been various finite element methods. Concretely, if (1.7) represents the curl-div first-order system (1.1), there are some methods that can work for singular solution, e.g., see [11], [12], [13], [14], [15], [16] and references therein. If (1.7) represents the curlcurl-div second-order system (1.2), there are also some methods available, e.g., see [17], [18], [19], [20], [21], [22] and references therein. All these methods actually do not take the original formulation (1.7), and instead, modifications are introduced on the div operator to weaken it to some extent, avoiding the original L2 integral form Ωdiv(ɛu)div(ɛv).

The key feature of the mixed approach (1.8) is that the div(divergence) operator is relaxed in the distributional sense through the multiplier variable p, and the solution u is sought in the larger space H0(curl;Ω). The relaxation is crucial for an efficient finite element discretization, and in particular, the well-known edge elements such as Nédélec elements have been thus popularly employed in computational electromagnetism. The edge elements, of course, can correctly and optimally numerically solve the singular solution (1.10). However, the challenge for (1.8) is due to the saddle-point structure, which may not be favorable for large-scale computations in practice. The preconditioning is particularly necessary for solving the saddle-point system and is still in progress (cf. [23], [24] and references therein). Several mixed methods based on (1.8) can be found in [6], [8], [9], [25], [26], [27], [28], [29], etc.

The perturbation approach (1.9) is indeed simple and is coercive. It is closely related to the mixed approach (1.8). On the one hand, the theory for both are almost the same; on the other hand, for the system with div(ɛu)=g0, (1.9) is a two-stage approach, where a scalar variable needs to solve in advance to transform div(ɛu)=g0 into a divergence-free equation, see [26]. The challenge for (1.9) is that the coercivity degenerates as δ00 and the so-called consistency or error orthogonality of the finite element solution does not hold. In some circumstances, these issues are not favorable for adaptive algorithms, i.e., only very fine initial meshes can secure the convergence and optimality [30]. To efficiently solve (1.9), although it is coercive, some sophisticated preconditionings are needed in order to obtain the δ0-uniform convergence (e.g., cf. [10] and references therein).

In this paper, we shall propose a new unifying approach for (1.3). The approach is a mixed variational formulation. In addition to the solution u itself, we find an additional variable, called pseudo electric displacement, denoted by w. This additional variable is quite close to the electric displacement variable D=ɛu(neglecting the electric polarization effect). Precisely, the main part in the bilinear form we propose reads as follows: Ω(ɛ1wu)(zɛv)+Ωμ1curlucurlv+Ωdivwdivz.Now, u is relaxed from the space H0(curl;Ω)H(div;ɛ;Ω) to the space H0(curl;Ω) only, like (1.8), (1.9), and can be efficiently numerically solved by the well-known Nédélec elements which are H(curl;Ω)-conforming. The additional variable w, mainly representing ɛu, is solved in H(div;Ω), and the well-known Raviart–Thomas–Nédélec elements which are H(div;Ω)-conforming are used for approximations. Roughly speaking, the idea behind (1.11) is motivated by the fact that uH0(curl;Ω) while its divergence div(ɛu)L2(Ω), i.e., ɛuH(div;Ω). It is therefore reasonable to pursue the approximation of u by independent tangential-oriented approximation keeping tangential-continuity of n×u and normal-oriented approximation keeping the normal-continuity (ɛu)n=wn. Thus, from (1.11), we can obtain H0(curl;Ω) approximation of u and can obtain H(div;Ω) approximation of ɛu through the approximation of the variable w. Although the approach (1.11) computes an additional variable w, this variable has its physical meaning; for the usual Maxwell problem (1.2) and the corresponding Maxwell eigenproblem, w exactly represents the variable ɛu. Such variable is none other than the electric displacement without the polarization effect. In electromagnetism, the electric field and the electric displacement field are all of physical interest. Not restricted to the stationary system (1.3), the approach (1.11) also covers the time-harmonic problem (1.5) and the eigenproblem (1.6) (see Section 4).

In comparison with (1.7), (1.8), (1.9) in the finite element discretizations, the new approach (1.11) has several advantages. There is no need for modifications at the discrete level, allowing conforming discretizations, i.e., both the continuous and discrete problems share the same variational formulation. The same coercivity holds in both continuous and discrete problems. The approach (1.11) yields a symmetric, positive definite(SPD) algebraic system. The coercivity is parameter-free and the error orthogonality holds, unlike (1.9). Classical efficient solvers and preconditioning techniques for SPD system are readily applicable, and (1.11) is more desirable in practical large-scale computations. What is more, we can simultaneously obtain L2 curl-convergent approximation and L2 div-convergent approximation for the exact solution, although both approximations come respectively from H0(curl;Ω) and H(div;Ω). Consider the cases of the systems (1.1), (1.2), where g1=g2=0, i.e., the divergence-free constraints hold, the so-called Coulomb Gauge condition, well-known as Gauss law in computational electromagnetism. In these cases, as will be shown that w=ɛu, we have uuh0+curl(uuh)0=O(hs), ɛuwh0+div(ɛu)divwh0=O(hs),and in general, although wɛu, there always holds the L2 div-convergence(see Remark 4.3) div(ɛu)divwh0=O(hs).On the contrary, the previous three approaches (1.7), (1.8), and (1.9) cannot produce such L2 div-convergent approximation. The L2 div-convergence would be useful for the problems where the so-called Gauss law (the divergence equation) is of interest.

We shall study the well-posedness of the new approach whose main part is (1.11), including the existence and uniqueness of the solution (u,w). We shall also study the relationship of (1.11) to the generalized Maxwell problem (1.3). The relationship is not immediately clear in their appearances. This is particularly the case when adopting different test functions in (1.11). For that purpose, we additionally consider the well-posedness of the displacement-based formulation whose main part is (1.7). All these will be done in the general settings: the domain Ω is multiply connected, the domain boundary Γ has a number of connected components, and the media occupying Ω are discontinuous, anisotropic and inhomogeneous. More importantly, we establish the coercivity on H0(curl;Ω)×H(div;Ω) for (1.11).

We shall then propose the finite element method of the new approach (1.11). Since we are interested in the singular solution as defined in (1.10), we only analyze the lowest-order Nédélec element of first kind for approximating u and the lowest-order Raviart–Thomas–Nédélec element for approximating w. In particular, to deal with the Nédélec element for interpolating the solution with a low regularity, we establish a regular-singular decomposition, in order to obtain the error bounds for the regularity 0<s1/2 from the canonical interpolation in discontinuous, anisotropic and inhomogeneous media. Although the solution lives in the heterogeneous media, but this decomposition is independent of the media. Actually, we show this decomposition using the well-known global embedding (5.8) and the L2-orthogonal decomposition for homogeneous media and Lipschitz polyhedron (see the proof of Lemma 5.1). Of course, for homogeneous media, the so-called smoothed projection in [31], [32], [33], [34] may also be used for constructing an interpolation for the low regularity solution. Thus, for all 0<s1, we obtain the convergence rate between the finite element solution and the exact solution. For s=0, thanks to the coercivity and the error orthogonality of the new approach, the quasi-optimal error estimates in the norms of H0(curl;Ω)×H(div;Ω) trivially follow, and so does the convergence.

The remaining part of this paper is arranged as follows. In Section 2, we state the generalized Maxwell problem and review the L2-orthogonal decomposition. In Section 3, we study the well-posedness of both the generalized Maxwell problem and the displacement-based variational problem and the relationship between the two problems. In Section 4, we propose the new mixed variational problem and study its well-posedness and the relationship to the problems in the previous section. In Section 5, we give and analyze the finite element method of the mixed problem. In Section 6, we report the numerical results. And finally in Section 7, a conclusion is given.

Section snippets

A generalized Maxwell problem

In this subsection, we state the generalized Maxwell problem.

Given a Lipschitz polyhedron ΩR3, with boundary Γ. Let ω=(ωij) be some symmetric real tensor function in R3×3, satisfying ωij=ωjiL(Ω), 1i,j3, and the ellipticity property 1i,j3ωijξiξjC1i3ξi2a.e. in Ω ,ξ=(ξi)R3 for some positive constant C. Let ɛ,μR3×3 be such two tensors. Without loss of generality, we assume that ɛ and μ are given by piecewise Lipschitz continuous functions, i.e, there is a partition P={Ωl;1lL},

Well-posedness and variational problem

In this section, we study the well-posedness of the generalized Maxwell problem (2.1), and study a variational problem of it.

Mixed variational formulation

In this section, we propose a mixed variational formulation and study its well-posedness and the relationship to the generalized Maxwell system (2.1). The mixed variational formulation caters to different test functions.

Set ((u,w),(v,z))(ɛ1wu,zɛv)+(μ1curlu,curlv)+(divw,divz)+i=2n1|Γi|ΓiwnΓizn.

Here we consider a mixed variational problem.

To find uH0(curl;Ω) and wH(div;Ω) such that ((u,w),(v,z))=(ɛ1f,z)for all vH0(curl;Ω) and zH(div;Ω).

Very interesting, we can use different test

The finite element method

In this section, we consider the finite element discretization.

For general purpose, we consider the following continuous variational problem.

Find (u,w)H0(curl;Ω)×H(div;Ω) such that ((u,w),(v,z))=(χ,(v,z))(v,z)H0(curl;Ω)×H(div;Ω),where (χ,(v,z)) represents the right-hand side of (4.2) or that of (4.3). Let UhH0(curl;Ω) and WhH(div;Ω) be two finite element subspaces. The corresponding finite element method is to find (uh,wh)Uh×Wh such that ((uh,wh),(vh,zh))=(χ,(vh,zh))(vh,zh)Uh×Wh.

Numerical results

In this section, we report the numerical results for the new mixed method (4.2) of this paper. We consider the two-dimensional Maxwell problem (4.6) and the Maxwell eigenproblem (4.9). We use the lowest-order Nédélec element and the lowest-order Raviart–Thomas element on the triangle mesh.

Conclusion

We have presented and analyzed a mixed variational formulation in terms of the electric field and the pseudo electric displacement field for the solution of the generalized Maxwell problem. We have also presented the finite element method with the use of the edge element and flux element for the approximations in H0(curl;Ω)×H(div;Ω). We have established the well-posedness of both continuous and discrete problems by mainly proving the coercivity, and the error bounds for singular solutions. L2

Acknowledgments

The authors would like to thank the anonymous referees’ suggestions on the improvement of the presentation of this paper. The first author was supported by National Natural Science Foundation of China (11971366, 11571266 and 11661161017) and by the Hubei Key Laboratory of Computational Science, Wuhan University, China (2019CFA007).

References (51)

  • GiraultV. et al.

    Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms

    (1986)
  • KikuchiF.

    Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism

    Comput. Methods Appl. Mech. Engrg.

    (1987)
  • BespalovA.N.

    Finite element method for the eigenmode problem of a RF cavity resonator

    Soviet J. Numer. Anal. Math. Model.

    (1988)
  • ReitzingerS. et al.

    An algebraic multigrid method for finite element discretizations with edge elements

    Numer. Linear Algebra Appl.

    (2002)
  • BrambleJ.H. et al.

    A new approximation technique for div–curl systems

    Math. Comp.

    (2003)
  • BensowR. et al.

    Discontinuous least-squares finite element method for the div–curl problem

    Numer. Math.

    (2005)
  • DuanH.Y. et al.

    A least squares finite element method for the magnetostatic problem in a multiply-connected Lipschitz domain

    SIAM J. Numer. Anal.

    (2007)
  • LeeE.J. et al.

    Weighted-norm first-order system least-squares (FOSLS) for div/curl systems with three dimensional edge singularities

    SIAM J. Numer. Anal.

    (2008)
  • DuanH.Y. et al.

    A finite element method for a curlcurl-graddiv eigenvalue interface problem

    SIAM J. Numer. Anal.

    (2016)
  • DuanH.Y. et al.

    Analysis of a continuous finite element method for H(curl, div)-elliptic interface problem

    Numer. Math.

    (2013)
  • LeeE.J. et al.

    FOSLL method for the eddy current problem with three-dimensional edge singularities

    SIAM J. Numer. Anal.

    (2007)
  • DuanH.Y. et al.

    The local L2 projected C0 finite element method for Maxwell problem

    SIAM J. Numer. Anal.

    (2009)
  • BrambleJ.H. et al.

    A least-squares approximation method for the time-harmonic maxwell equations

    J. Numer. Math.

    (2005)
  • DuanH.Y. et al.

    C0 Elements for generalized indefinite Maxwell’s equations

    Numer. Math.

    (2012)
  • WathenM. et al.

    Preconditioners for mixed finite element discretizations of incompressible MHD equations

    SIAM J. Sci. Comput.

    (2017)
  • Cited by (2)

    View full text