Numerische Mathematik

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1 Numer. Math. (2012) 122:61 99 DOI /s x Numerische Mathematik C 0 elements for generalized indefinite Maxwell equations Huoyuan Duan Ping Lin Roger C. E. Tan Received: 31 July 2010 / Revised: 16 August 2011 / Published online: 24 February 2012 Springer-Verlag 2012 Abstract In this paper we develop the C 0 finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H r regularity for some r < 1. The ingredients of our method are that two mass-lumping L 2 projectors are applied to curl and div operators in the problem and that C 0 linear element or isoparametric bilinear element enriched with one elementbubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C 0 Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H r regularity where r may vary in the interval [0, 1), we obtain the error bound O(h r ) in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds. H. Duan (B) School of Mathematical Sciences, Nankai University, Tianjin , People s Republic of China hyduan@nankai.edu.cn P. Lin Division of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, UK plin@maths.dundee.ac.uk R. C. E. Tan Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore , Singapore scitance@nus.edu.sg

2 62 H. Duan et al. Mathematics Subject Classification (2000) 65N30 35Q61 35J47 65N15 65N12 65M60 1 Introduction In this paper we shall develop the C 0 finite element method for a generalized indefinite Maxwell problem with a nonsmooth solution (i.e., the solution does not have the H 1 regularity). We consider a simply-connected Lipschitz domain R 2 with boundary Ɣ and let u = (u 1, u 2 ) denote the unknown field. Then, the generalized indefinite Maxwell problem that we shall consider is the following curlcurl-grad div indefinite problem which reads: find u such that curl curl u div u λ u = f in (1.1) for a given f (L 2 ( )) 2 together with some suitable boundary conditions. Here λ is a real number, and for vector-valued u = (u 1, u 2 ), the curl operator curl u = x u 2 y u 1 and the div operator div u = x u 1 + y u 2, and for scalar φ, the curl operator curl φ = ( y φ, x φ) and the gradient operator φ = ( x φ, y φ). With the same curl/div bilinear form in (1.5) below, the above system covers several important examples in electromagnetism and fluid-structure interaction problems [8,37]: Case 1 The time-harmonic Maxwell equations with λ>0, i.e. curl curl u λ u = J, div u = g. Here and below J and g are given functions. Case 2 The vector potential equations: curl curl u = J, div u = g. Case 3 The time-discretization of the transient Maxwell equations with the parameter λ >0 being inversely proportional to the time step: curl curl u λ u = J, div u = g. Case 4 The curl-div magnetostatics problem: curl u = J, div u = g. Case 5 The fluid structure interaction problem: div u = J, curl u = g.

3 C 0 elements for generalized indefinite Maxwell equations 63 As usual, we may consider two kinds of boundary conditions for (1.1). Let n be the unit outward normal vector to Ɣ and τ the unit tangential vector along Ɣ. The first kind of boundary conditions are u τ = 0, div u = 0 on Ɣ. (1.2) Note that in Cases 1 4, the boundary condition on div u is not needed. The second kind of boundary conditions are u n = 0, curl u = 0, on Ɣ. (1.3) Note that in Cases 4 5 the boundary condition on curl u is not needed. In this paper, we shall develop the C 0 finite element method for problem (1.1) and (1.2). The adjustment of the method to problem (1.1) and (1.3) is straightforward. Let us now introduce a Hilbert space U ={v (L 2 ( )) 2 : curl v L 2 ( ), div v L 2 ( ), v τ Ɣ = 0}. (1.4) Problem (1.1) and (1.2) can be stated as a variational problem with the primal formulation as follows: Find u U such that (curl u, curl v) + (div u, div v) λ(u,v)= ( f,v) v U, (1.5) where (, ) denotes the L 2 inner product. When considering numerical methods for (1.5), it is desirably natural to use the C 0 Lagrangian finite element method depicted in [14], since both H(curl; ) H(div; )-conforming piecewise polynomial finite elements are necessarily in (H 1 ( )) 2 and continuous, and since problem (1.5) is U-elliptic due to the fact that the curl/div part of the bilinear form induces an equivalent norm over U (cf. [6,26]), and since numerous softwares and algorithms are nowadays available in the literature for C 0 Lagrange elements (cf. [32,33,35] and references therein). Unfortunately, whenever the domain is nonsmooth with reentrant corners (e.g., corners in nonconvex polygon), the use of C 0 Lagrange elements is challenging for the numerical solution of problem (1.1) due to the well-known wrong convergence phenomenon. For nonsmooth domains the solution of (1.1) has the H r regularity only for some r < 1, see [3,16,19], even if the right-hand-side is smooth. We refer such low regularity solution to as nonsmooth solution (otherwise, if the solution has the H 1 regularity, we call it smooth solution). In that case the C 0 finite element solution from (1.5) converges to some element in (H 1 ( )) 2 but not to the true solution that is only in (H r ( )) 2 with r < 1 a Hilbert space much larger than (H 1 ( )) 2 (see [7,18,28 30]). Meanwhile, as is now well-known, it is the curl/div bilinear form in (1.5) that accounts for this wrong convergence (see [17,19 21,23]). This fact highlights that the way out for the C 0 elements to accommodate the nonsmooth solution is to modify the curl/div part of the bilinear form in (1.5). We should as well point out that all the existing finite element methods (C 0 -element method, edge element method [30,36], discontinuous Galerkin method [11,31], etc) do not work for nonsmooth solutions if

4 64 H. Duan et al. no modifications are made on the primal variational formulation (1.5) of the general curlcurl-grad div problem (1.1). Although the wrong convergence has been known for long in both mathematical and engineering communities, attempts have never been stopped in using C 0 finite element methods to approximate the nonsmooth solution due to the advantages of C 0 Lagrange elements in practice. Unexpectedly, only till recent years have several practically successful methods been available in the literature: the weighted method [18] forcase 1, theh 1 least-squares method [9] forcase 4, the weighted dualpotential least-squares method [34] for the first-order system of Maxwell equations in term of electric and magnetic fields, the weighted mixed method [13] for Case 1, and the L 2 projection method [23] forcase 2. The weighted and mixed methods consist of a weight function to the div operator where the weight function is defined according to the geometric singularities in the vicinity of reentrant corners and they require that the C 0 finite element space includes the gradient of some C 1 element. The H 1 least-squares method formulates Case 4 into the least-squares formulation under the H 1 inner products. The weighted dual-potential least-squares method introduces additional potential variables to formulate a dual first-order system of curl/div operators, places a weight function of geometric singularities to the div operator terms, and recovers the original solutions by a post-processing method. The L 2 projection method applies two element-local L 2 projectors to both curl and div operators for Case 2. The advantages of this method are that the L 2 projections are used in place of either the weight functions in the weighted method or the H 1 projections in the H 1 method and that the inclusion of the gradient of C 1 element is not required and no additional potential variables are introduced. In this paper, different from our previous work [23], a new L 2 projected C 0 finite element method is proposed and a new theory for stability and error estimates is developed, in addition to the same advantages (no weights of geometric singularities, no H 1 inner products, no gradient of C 1 elements, etc), with four main features as follows: (i) To deal with the general indefinite problem (1.1) and to develop a new theory for stability and error estimates. For practical purposes of Cases 1 5 it is of most interest to study the general problem (1.1), especially the indefinite case with λ>0. None of the aforementioned C 0 finite element methods has been applied to (1.1), and no theory of the associated stability and error estimates is available for the indefinite problem. As a rule, the duality argument is fundamental in dealing with the indefinite elliptic problem [10]. But, for the vectorial elliptic problem (1.1), its solution may not have the H 1 regularity. Thus, a direct application of the usual duality argument seems to be impossible. It turns out that the handling of the error estimates issue in the indefinite case of problem (1.1) whose solution is nonsmooth is very technical. We instead develop a completely different theory for stability and error estimates. This new theory comprises two Fortin-type interpolations and one Inf-Sup condition (using terminologies in classical mixed finite element method [12]). We then elaborate an ad hoc duality argument for the error estimates of the indefinite problem with a nonsmooth solution.

5 C 0 elements for generalized indefinite Maxwell equations 65 (ii) To design a simpler finite element scheme without placing the mesh-dependent stabilization term in our previous work where the stabilization therein is for the stability. Through our new theory such stabilization is unnecessary in the case of λ = 0. Hence, this may greatly facilitate the adaptation of the L 2 projection method to other more complex problems such as the interface problem in discontinuous, anisotropic and nonhomogeneous media. (iii) To have fewer degrees of freedom using linear or bilinear elements enriched with some element-bubbles, 8(= 4 2) degrees of freedom on each triangle or 10(= 5 2) degrees of freedom on each quadrilateral. After the static condensation of the element bubbles of the C 0 finite element space of the solution, one obtains a three-nodes or four-nodes C 0 element method. The number of degrees of freedom is less than half of that in the other methods. (iv) To allow for less regular solution in (H r ( )) 2 even for 0 r 1/2 aswell as r > 1/2. The exact solution is required to have the H r regularity with r > 1/2 elsewhere whenever the finite element interpolation error estimates on element edges/faces are involved with. For Lipschitz polygonal domains we indeed have r > 1/2. However, for general Lipschitz domains, the exact solution may have the H 1/2 regularity only and there are also some cases (such as non-lipschitz domains) where the regularity of the solution would be worse, i.e., r < 1/2. To work for any 0 r 1 and to use simpler C 0 elements we adopt two new L 2 projectors for the div and curl operators, mass-lumping L 2 projectors, which are defined according to the mass-lumping L 2 inner products (, ) 0,h which is the discrete version of the L 2 inner product (, ) (see Sect. 2). Now let R h and R h denote the two mass-lumping L 2 projectors, respectively, for div and curl operators and let U h be the C 0 finite element space of the solution. The finite element method we propose for problem (1.1) and (1.2) reads: find u h U h such that for all v h U h L h (u h,v h ) := (R h (curl u h ), R h (curl v h )) 0,h + ( R h (div u h ), R h (div v h )) 0,h λ(u h,v h ) = ( f,v h ). (1.6) Note that the mass-lumping L 2 projections in (1.6) can be easily implemented since the associated mass-lumping L 2 inner products (, ) 0,h generate diagonal matrices. For each component of the nonsmooth solution we employ the C 0 linear element enriched with one element-bubble for each triangle or the C 0 isoparametric bilinear element enriched two element-bubbles for each quadrilateral. The element-bubbles can be statically eliminated at element levels. So (1.6) may be viewed as a C 0 threenodes (linear) element method for triangles, or a C 0 four-nodes (bilinear) element method for quadrilaterals. We obtain the optimal error estimates O(h r ) in an energy norm between the exact solution with the H r regularity and the finite element solution of (1.6). To the authors knowledge, the method (1.6) is the first C 0 finite element method proposed for the general problem (1.1) and (1.2) with a nonsmooth solution and with the least degrees of freedom. The rest of this paper is arranged as follows. In Sect. 2, we describe the finite element method. In Sect. 3, we list theoretical results for curl/div operators. Section 4

6 66 H. Duan et al. is devoted to the establishment of Inf-Sup inequality and the Fortin-type interpolations. In Sect. 5 we obtain coercivity and error bounds in an energy norm. In Sect. 6 we report the numerical results to illustrate the theoretical error bounds. Concluding remarks are given in the last section. 2 The finite element method In this section we define the finite element method (1.6) in details. For ease of exposition, let be the polygonal domain with a piecewise straightsided boundary. Denote by T h the conforming triangulation of into shape-regular triangles or quadrilaterals [10,14], where h = max K Th h K and h K is the diameter of K. We denote by ˆK the reference triangle element in the (ξ, η) space with vertices (0, 0), (0, 1), (1, 0), or the reference square element ˆK =[ 1, 1] [ 1, 1]. Associated with each K T h, there exists exactly one invertible mapping F K : ˆK K, such that K = F K ( ˆK ). TheinverseofF K is denoted by F 1 K. Denote by P 1( ˆK ) the space of linear polynomials on the reference triangle element and Q 1 ( ˆK ) the space of bilinear polynomials on the reference square element. Put P 1 (K ) = P 1 ( ˆK ) F 1 K, Q 1(K ) = Q 1 ( ˆK ) F 1 K. Introduce the element-bubble b K = b ˆK F 1 K, where b ˆK is the bubble on ˆK, b ˆK = ξη(1 ξ η) for triangle; b ˆK = (1 ξ 2 )(1 η 2 ) for square. We use N 1 (x, y) Q 1 (K ) to denote the local basis function associated with the vertex (1, 1) of ˆK. Introduce the usual Hilbert spaces [1]: H 1 ( ) ={q L 2 ( ) : q (L 2 ( )) 2 }, H 1 0 ( ) ={q H 1 ( ) : q Ɣ = 0}, H 1 ( )/R ={q H 1 ( ) : q = 0}.Also,H 0(curl; ) ={v (L 2 ( )) 2 : curl v L 2 ( ), v τ Ɣ = 0}. On K T h, we introduce the element space D(K ) := { span{(0, 1), (1, 0)} on triangles, span{(0, 1), (1, 0), curl N 1, N 1 } on quadrilaterals, (2.1) and the element-bubble space B(K ) := b K D(K ), (2.2) and B h ={v (H 1 0 ( ))2 : v K B(K ), K T h }. (2.3) Denote by R 1 (K ) the linear or isoparametric bilinear element space on K T h as follows: { P1 (K ) on triangles, R 1 (K ) := Q 1 (K ) on quadrilaterals, (2.4) and introduce the C 0 finite element space V h ={q H 1 ( ) : q K R 1 (K ), K T h }. (2.5)

7 C 0 elements for generalized indefinite Maxwell equations 67 We define the C 0 finite element space U h for the solution of problem (1.1) and (1.2) by U h = (V h ) 2 H 0 (curl; ) + B h (2.6) and the C 0 finite element spaces Q h and W h for the two L 2 projections R h and R h, respectively, by Over V h,weuse Q h = V h H0 1 ( ), (2.7) W h = V h H 1 ( )/R. (2.8) (, ) 0,h (2.9) to denote the discrete L 2 inner product that is chosen as an approximate of the L 2 inner product (, ). In Remark 2.1 below, the concrete (, ) 0,h is given. Remark 2.1 For triangles, over V h we choose the discrete L 2 inner product (, ) 0,h as follows: (p, q) 0,h = K 3 K T h 3 p(a i )q(a i ), (2.10) where K is the area of K, and a i, 1 i 3, the vertices of K. This discrete L 2 inner product (p, q) 0,h is the so-called mass-lumping approximate of the L 2 inner product (p, q) = pq,see[41]. Note that the matrix generated from (2.10) is diagonal. For quadrilaterals, we may choose (, ) 0,h as the four-node quadrature rule [25]. For any given v (L 2 ( )) 2, we define R h (div v) Q h and R h (curl v) W h as follows: i=1 ( R h (div v),q) 0,h = (v, q) q Q h, (2.11) (R h (curl v), w) 0,h = (v, curl w) w W h. (2.12) The finite element method we propose is stated as (1.6), i.e., to find u h U h such that where L h (u h,v h ) = ( f,v h ) v h U h, (2.13) L h (u h,v h ) = (R h (curl u h ), R h (curl v h )) 0,h + ( R h (div u h ), R h (div v h )) 0,h λ(u h,v h ). (2.14)

8 68 H. Duan et al. Remark 2.2 Note that R h (div v) and R h (curl v) are not really L 2 projections of div v and curl v, since, (1) the inner products are different from both sides; (2) v does not satisfy v τ = 0in(2.12); (3) div v and curl v are not necessarily L 2 functions. On the other hand, due to the fact that (, ) 0,h is the approximation of the L 2 inner product (, ), whenever curl v L 2 ( ) with v satisfying v τ Ɣ = 0 and for any v with div v L 2 ( ), R h (curl v) and R h (div v) maybeviewedasl 2 projections. This is the case for any finite element function v U h H 0 (curl; ) H(div; ). For this reason, we would call R h and R h mass-lumping L 2 projections. Here mass-lumping means to emphasize that the definitions of R h and R h are based on the discrete L 2 inner product (, ) 0,h. For convenience, R h and R h will be simply referred to as L 2 projections [but the real matter refers to (2.11) and (2.12)]. Remark 2.3 Since the element-bubble in U h can be statically eliminated at element levels, so (2.13) is in essence C 0 linear or bilinear finite element method. Remark 2.4 From (2.11) and (2.12) we have seen that (2.13) is valid even if the solution has the H 0 = L 2 regularity only. It is then expected that for all 0 r 1, (2.13) can give a correctly convergent C 0 finite element solution at the rate which is optimal relative to the H r regularity of the exact solution of problem (1.1) and (1.2). Remark 2.5 If the solution is more regular, say belonging to H 1 and above, we may consider to employ higher-order C 0 elements. For example, let us consider triangle meshes. Let P l (K ) denote the space over K of polynomials of the total degree in both x and y coordinate variables not greater than l for the integer l 1. Defining V h := {q H 1 ( ) : q K P l (K ), K T h } and B h := {v (H0 1( ))2 : v K b K D(K ), K T h }, where D(K ) := (P l 1 (K )) 2, we may use (2.6), (2.7), (2.8) with these new V h and new B h here for approximating the solution and for defining R h and R h. Regarding quadrilaterals, we may have similar higher-order elements, only noting that D(K ) needs to contain the divergence and curl of Q l (K ) which is the space over K of isoparametric polynomials of the degree in each coordinate variable not greater than l, in order that the Inf-Sup inequality can be established. For example, for rectangles, we may choose D(K ) := Q l 1,l (K ) Q l,l 1 (K ), where Q i, j (K ) stands for the space over K of polynomials of the degree in x and y coordinate variable not greater than i and j, respectively. A ready choice is of course D(K ) := (Q l (K )) 2 for rectangles. 3 Preliminaries For the analysis of stability and error estimates, with the curl/div Hilbert spaces, we recall several fundamental functional analysis results for curl/div operators in this section. Let H 1 ( ), H 1 0 ( ), and H 1 ( )/R be equipped with norm 1 and semi-norm 1. We also need Hilbert space H s ( ) with norm s for s R, where H 0 ( ) = L 2 ( ). In addition, for div and curl Hilbert spaces as follows [26]: H(div; ) ={v (L 2 ( )) 2, div v L 2 ( )}, H(curl; ) ={v (L 2 ( )) 2, curl v L 2 ( )}

9 C 0 elements for generalized indefinite Maxwell equations 69 we equip them with norms 0;div and 0;curl, respectively: v 2 0;div = v div v 2 0, v 2 0;curl = v curl v 2 0. The Hilbert space H(curl; ) H(div; ) is equipped with the norm v 2 0;curl;div = v curl v div v 2 0. Proposition 3.1 [6,26] For any v H 0 (curl; ) H(div; ) we have curl v div v 2 0 C v 2 0. (3.1) As a result, the Hilbert space H 0 (curl; ) H(div; ) may be equipped with the following norm curl;div which is equivalent to the norm 0;curl;div, v 2 curl;div := curl v div v 2 0. Proposition 3.2 [2,6,26] We have the following L 2 orthogonal decomposition of vector fields: (L 2 ( )) 2 = curl (H 1 ( )/R) + H0 1 ( ). (3.2) Proposition 3.3 [36] For any given f (L 2 ( )) 2 and for any given λ 0 or for any given λ>0 which is not the eigenvalue of the curlcurl-grad div operator, there exists a unique solution u H 0 (curl; ) H(div; ) to problem (1.1) and (1.2), satisfying u 0;curl;div C f 0. (3.3) Throughout this paper, we will always assume that λ is not the eigenvalue of the curlcurl-grad div operator. Assumption A We require that the following continuously embedding holds for some r 0: H 0 (curl; ) H(div; ) (H r ( )) 2, (3.4) where for any v H 0 (curl; ) H(div; ) we have v r C v curl;div. (3.5) Note that for Lipschitz polygons with piecewise smooth boundary r > 1/2 and for general Lipschitz domains r = 1/2, see [2,16]. Nevertheless, for non-lipschitz domains, r may be less than 1/2. Throughout this paper we do not require r 1/2, although we have assumed a Lipschitz polygon.

10 70 H. Duan et al. Proposition 3.4 [2,24,26] For any given w L 2 ( )/R and p L 2 ( ), there exists a unique solution v H 0 (curl; ) H(div; ) to the following problem: curl v = w, div v = p in, v τ = 0 on Ɣ. (3.6) Moreover, v (H r ( )) 2 for some r 0, satisfying v r C ( w 0 + p 0 ). (3.7) Remark 3.1 In two dimensions, the H r regularity in Assumption A and in Proposition 3.4 come from the regularity H 1+r 1 and H 1+r 2 of both the solutions to the Poisson equation of Laplace operator for some right-hand-side g [27], respectively, φ = g (3.8) with Dirichlet and Neumann boundary condition. We may take r = min(r 1, r 2 ), where r i 0, i = 1, 2. Before closing this section, let us recall Green s formula of integration by parts (div v,q) 0,D + (v, q) 0,D = D (curl v, w) 0,D (v, curl w) 0,D = D v nq v H(div; D), q H 1 (D), v τw v H(curl; D), w H 1 (D), Here and in the subsequent, (, ) 0,D, 0,D denote the L 2 inner product and the L 2 norm on D, and when D =,(, ) 0, = (, ), 0, = 0. 4 The Fortin-type interpolation In this section we construct two Fortin-type interpolations and an Inf-Sup inequality, associated with the trilinear form over H(curl; ) H(div; ) L 2 ( ) L 2 ( ) as follows: b(v; (p,w))= (curl v, w) + (div v, p). (4.1) The Fortin-type interpolations are essential in the establishment of the error estimates between the exact solution of (1.1), (1.2) and the finite element solution of (2.13) when reasoning the duality argument for the indefinite problem for λ>0 in next section.

11 C 0 elements for generalized indefinite Maxwell equations The Fortin-type interpolation and the Inf-Sup inequality Introduce over U h v h 2 h = v h h 2 2r K K T h where 0 r 1 comes from Remark 3.1. div v h 2 0,K + Proposition 4.1 We have the following inclusions: K T h h 2 2r curl R 1 (K ) R 1 (K ) D(K ) K curl v h 2 0,K, (4.2) Proof For triangles, R 1 (K ) = P 1 (K ), D(K ) = (P 0 (K )) 2 the space of constants on K, the inclusion is obvious. For quadrilaterals, R 1 (K )= Q 1 (K ), D(K ) = span{(0, 1), (1, 0), curl N 1, N 1 }, the inclusion follows from Theorem 4.3 in [22]. Theorem 4.1 For any given v H 0 (curl; ) H(div; ) (H r ( )) 2, there is a Fortin-type interpolation ṽ U h of v such that b(ṽ; (p,w)) = b(v; (p,w)) p Q h, w W h, (4.3) ṽ h C v r. (4.4) Moreover, the following interpolation properties (4.5) (4.8) hold: (ṽ v) = 0 for triangles, for all K T h, (4.5) K or (ṽ v)q = 0 for all q D(K ) for quadrilaterals, for all K T h. (4.6) K v ṽ 0 Ch r v r, (4.7) 1/2 K v ṽ 2 0,K + ṽ r C v r. (4.8) K T h h 2r Proof We construct ṽ U h in the following way. We first let π h v (V h ) 2 H 0 (curl; ) U h be the interpolation of v, satisfying [4,5,10,15,26,39] v π h v 0 Ch r v r, (4.9) 1/2 K v π hv 2 0,K + π h v r C v r. (4.10) K T h h 2r

12 72 H. Duan et al. We then define ṽ U h as follows: ṽ(a) = π h v(a) for all vertices a in T h, (4.11) (ṽ v) = 0 for all triangles in T h (4.12) K or K (ṽ v)q = 0 for all q D(K ) for all quadrilaterals in T h. (4.13) It is not difficult to verify that ṽ U h is uniquely determined by (4.11) and (4.12) (for triangles) or (4.13) (for quadrilaterals). In fact, writing ṽ = v L + v B, with the linear part v L (V h ) 2 H 0 (curl; ) and with the bubble part v B B h, we see that the linear part v L is determined by (4.11) and the bubble part v B by (4.12) or(4.13) element-by-element (noting Proposition 4.1). In other words, K K v L = π h v, (4.14) v B = (v π h v) for triangles in T h, (4.15) K v B q = (v π h v)q q D(K ) for quadrilaterals in T h. (4.16) K On K T h, by the standard scaling argument [10,14,26] between K T h and the reference element ˆK, we can have from (4.15) or(4.16) that and we obtain v B 0,K C v π h v 0,K, (4.17) ṽ v 0,K C v π h v 0,K. (4.18) Hence, ṽ satisfies the same interpolation error properties (4.9) and (4.10)asπ h v, that is, (4.7) and (4.8) hold. Moreover, (4.5) and (4.6) come from the definitions (4.12) and (4.13), respectively. By the inverse estimates [10,14,26] over finite dimensional spaces h 1 r K v h 1,K C v h r,k v h U h, K T h, (4.19) we obtain from (4.8) that div ṽ 2 0,K + h 2 2r K K T h K T h h 2 2r K curl ṽ 2 0,K C ṽ 2 r C v 2 r, (4.20)

13 C 0 elements for generalized indefinite Maxwell equations 73 That is to say, (4.4) holds. On the other hand, we have from (4.12) or(4.13) and Proposition 4.1 b(ṽ v; (p,w))= (ṽ v, curl w p) = 0 for all p Q h,w W h,(4.21) and we have (4.3). Remark 4.1 We call such ṽ the Fortin-type interpolation which satisfies (4.3). Interms of L 2 projectors R h and R h, we see that (4.3) can be expressed as follows: R h (curl (v ṽ)) = 0, R h (div (v ṽ)) = 0. (4.22) In addition, under Assumption A with v r C v curl;div, we can then use v curl;div to replace v r in all the estimates in Theorem 4.1. Corollary 4.1 We have the following Inf-Sup inequality: b(v h ; (p h,w h )) sup C( p h 0 + w h 0 ) for all p h Q h, for all w h W h. v h U h v h h (4.23) Proof Consider the problem: Given w h W h, p h Q h, to find v H 0 (curl; ) H(div; ) such that curl v = w h, div v = p h v τ Ɣ = 0. (4.24) From Proposition 3.4 we know that the above problem has a unique solution v. Moreover, v is in (H r ( )) 2, satisfying We also have v r C( p h 0 + w h 0 ). (4.25) b(v ; (p h,w h )) = p h w h 2 0. (4.26) Let v h U h be the Fortin-type interpolant to v, as constructed in Theorem 4.1. We have b(vh ; (p h,w h )) = b(vh v ; (p h,w h )) + b(v ; (p h,w h )) = b(v ; (p h,w h )) = p h w h 2 0, (4.27) vh h C v r C( p h 0 + w h 0 ). (4.28)

14 74 H. Duan et al. Therefore, b(v h ; (p h,w h )) sup v h U h b(v h ; (p h,w h )) v h h vh h The proof is finished. = p h w h 2 0 v h h C( p h 0 + w h 0 ). 4.2 Properties of the kernel of b We shall investigate the properties of the kernel set of the trilinear form b, i.e., K h (b) [see (4.29) below], so that we can state the Inf-Sup condition (4.23) and the Fortintype interpolation in the subspace K h (b),thel 2 - orthogonal complement of K h (b) in U h. Some of the properties established for K h (b) will be also used in the error estimates in Sect. 5. Let K h (b) ={v U h : b(v; (p,w))= 0, p Q h, w W h }. (4.29) For the finite dimensional space U h we can have the following orthogonal decomposition with respect to the L 2 inner product (, ): satisfying U h = K h (b) + K h (b), (4.30) (u,v)= 0 u K h (b), v K h (b). (4.31) Remark 4.2 The kernel set K h (b) of b can be also expressed in terms of L 2 projectors R h and R h as follows: K h (b) ={v U h : R h (curl v) = 0, R h (div v) = 0}. (4.32) Proposition 4.2 For any v H(div; ) (H r ( )) 2, we have the following regularsingular decomposition: v = curl A + B, (4.33) where A H 1 ( )/R H 1+r ( ), B H 1 0 ( ) H 1+r ( ), satisfy A 1+r + B 1+r C( v r + v 0;div ). (4.34) Proof From Proposition 3.2 we have the following L 2 orthogonal decomposition: v = curl A + B, (4.35)

15 C 0 elements for generalized indefinite Maxwell equations 75 with A H 1 ( )/R, B H0 1 ( ), satisfying v 2 0 = A B 2 1, (4.36) where by Poincaré Friedrichs inequalities we have A 0 + B 0 C( A 1 + B 1 ) C v 0. (4.37) Since B satisfies Poisson equation B = div v in, B = 0 on Ɣ, (4.38) we know from Remark 3.1 that B H 1+r ( ) for some r 0, satisfying B 1+r C div v 0. (4.39) Since curl A 0 = A 1 in two-dimensions, it immediately follows from curl A = v B that A H 1+r ( ), which satisfies A 1+r v r + B 1+r C( v r + div v 0 ), (4.40) which, together with (4.37) and (4.39), leads to (4.34). Remark 4.3 If v H 0 (curl; ) H(div; ), then under Assumption A we have from (4.34) and Proposition 3.1 that A 1+r + B 1+r C v curl;div. (4.41) Proposition 4.3 For any v 0,h K h (b) and for any p H 1 0 ( ) H 1+r ( ) and w H 1 ( )/R H 1+r ( ), we have b(v 0,h ; (p,w)) Ch r v 0,h 0 ( p 1+r + w 1+r ). (4.42) Proof We know that there are p h Q h,w h W h, corresponding to p H 1 0 ( ) H 1+r ( ) and w H 1 ( )/R H 1+r ( ), satisfying [10,14,26] Since p p h 1 Ch r p 1+r, (4.43) w w h 1 Ch r w 1+r. (4.44) b(v 0,h ; (p h,w h )) = 0 p h Q h,w h W h, (4.45)

16 76 H. Duan et al. we have b(v 0,h ; (p,w)) = b(v 0,h ; (p p h,w w h )) = (curl v 0,h,w w h ) + (div v 0,h, p p h ) = (v 0,h, curl (w w h ) (p p h )) v 0,h 0 ( w w h 1 + p p h 1 ) Ch r v 0,h 0 ( p 1+r + w 1+r ). (4.46) Lemma 4.1 Let v (H r ( )) 2 H(div; ). Then, for all v 0,h K h (b), (v, v 0,h ) Ch r ( v r + v 0;div ) v 0,h 0. (4.47) Proof According to Proposition 4.2, we write v as follows: v = curl A B, with A H 1 ( )/R H 1+r ( ), B H0 1( ) H 1+r ( ), satisfying A 1+r + B 1+r C( v r + v 0;div ). We have (v, v 0,h ) = (curl A B,v 0,h ) = (A, curl v 0,h ) + (B, div v 0,h ) = b(v 0,h ; (B, A)), and we therefore have from Proposition 4.3 that (v, v 0,h ) = b(v 0,h ; (B, A)) Ch r v 0,h 0 ( A 1+r + B 1+r ) Ch r ( v r + v 0;div ) v 0,h 0. (4.48) Corollary 4.2 Let ṽ = ṽ 0 + ṽ0, with ṽ 0 K h (b), ṽ0 K h(b), be the Fortin-type interpolation of v H 0 (curl; ) H(div; ) (H r ( )) 2. Then, under Assumption A, ṽ 0 0 Ch r v curl;div. (4.49) Proof Since ṽ = (ṽ 0, ṽ 0 ) = (ṽ, ṽ 0 ) = (ṽ v, ṽ 0 ) + (v, ṽ 0 ), under Assumption A we have from Lemma 4.1 and Theorem 4.1 and Remark 4.1 that ṽ ṽ v 0 ṽ Ch r v curl;div ṽ 0 0 Ch r v curl;div ṽ 0 0.

17 C 0 elements for generalized indefinite Maxwell equations 77 Proposition 4.4 There is at least v h = 0 in K h (b). Proof Let ṽ be the Fortin-type interpolant to v H 0 (curl; ) H(div; ), where v = 0. Let ṽ = ṽ 0 + ṽ 0, where ṽ 0 K h (b), ṽ0 K h(b), then v h := ṽ0 K h (b). In fact, from Remark 4.1 we have seen that = 0 is the desired function in R h (curl (v ṽ)) = 0, R h (div (v ṽ)) = 0. Since ṽ 0 K h (b), from Remark 4.2 we have If ṽ 0 = 0, then we had That is to say, R h (curl (v ṽ0 )) = 0, R h (div (v ṽ0 )) = 0. R h (curl v) = 0, R h (div v) = 0. (div v, p h ) = 0 for all p h Q h, (curl v, w h ) = 0 for all w h W h. For any p H 1 0 ( ), there is a sequence {p h : p h Q h } h>0 such that [14] we have which implies Similarly, Hence, we have 0 = (div v, p h ) (div v, p) as h 0, (div v, p) = 0, div v = 0. curl v = 0. div v = 0, curl v = 0 in, v τ = 0 on Ɣ. It follows from Proposition 3.4 that v = 0. This contradicts v = 0.

18 78 H. Duan et al. Remark 4.4 We first see that the Inf-Sup inequality (4.23) in fact holds over K h (b), i.e., b(v h ; (p h,w h )) sup C( p h 0 + w h 0 ) for all p h Q h, v h K h (b) v h h for all w h W h. (4.50) Moreover, the Fortin-type interpolation in Theorem 4.1 of v H 0 (curl; ) H(div; ) (H r ( )) 2 can be taken in K h (b) only, i.e., the part ṽ0 of the L 2 -orthogonal decomposition of ṽ We clearly have ṽ = ṽ 0 + ṽ 0, with ṽ 0 K h (b) and ṽ 0 K h(b). (4.51) b(ṽ 0 ; (p h,w h )) = b(v; (p h,w h )) p h Q h, w h W h, (4.52) which can be also expressed in terms of L 2 projectors R h and R h as follows: R h (curl (v ṽ0 )) = 0, R h (div (v ṽ0 )) = 0. (4.53) Under Assumption A, from (4.49), (4.7) and Remark 4.1 we see that ṽ 0 satisfies v ṽ 0 0 v ṽ 0 + ṽ 0 0 Ch r v curl;div. (4.54) By inverse estimates on finite dimensional space U h h r v h r C v h 0 v h U h, (4.55) we have from (4.8), (4.49) and Remark 4.1 that ṽ0 r ṽ r + ṽ 0 r C v curl;div + h r ṽ 0 0 C v curl;div, (4.56) and we further have ṽ0 h C v curl,div, (4.57) 1/2 K v ṽ 0 2 0,K C v curl;div. (4.58) K T h h 2r Thus, we have seen that the ṽ0 satisfies the same properties as ṽ, except that v curl;div is in place of v r and that (4.5) and (4.6) do not hold for such ṽ0. Note that (4.5) or (4.6) are used to make (4.3) hold. The above estimates in terms of v curl;div (not v r ) are sufficient for the subsequent.

19 C 0 elements for generalized indefinite Maxwell equations The dual Fortin-type interpolation In what follows, we show that there is a dual Fortin-type interpolation over Q h W h for the trilinear form b. We consider the problem: Given p L 2 ( ) and w L 2 ( )/R, to find p Q h and w W h such that b(v h ; ( p, w)) = b(v h ; (p,w)) v h K h (b), (4.59) The choice of K h (b), together with the established Inf-Sup inequality (4.50), ensures that the stated problem has a unique solution. Over Q h W h, we introduce (p,w) 2,h := p w h 2r 2 K ( p 2 0,K + w 2 0,K ), (4.60) K T h where r is the same as in (4.2). With respect to h in (4.2) and,h in (4.60), by Cauchy Schwarz inequality, we have the boundedness of the trilinear form b as follows: b(v; (p,w)) C v h (p,w),h. (4.61) Theorem 4.2 Let p H 1 0 ( ), w H 1 ( )/R, and let ( p, w) Q h W h be constructed as in problem (4.59). We have p p 0 + w w 0 C inf (p q h,w z h ),h. (4.62) q h Q h,z h W h Proof Let q h Q h and z h W h. From the Inf-Sup inequality (4.50) wehave where p q h 0 + w z h 0 b(v h ; ( p q h, w z h )) sup, v h K h (b) v h h b(v h ; ( p q h, w z h )) = b(v h ; ( p p, w w)) + b(v h ; (p q h,w z h )) = b(v h ; (p q h,w z h )) C v h h (p q h,w z h ),h. We have By triangle-inequality to get p q h 0 + w z h 0 C (p q h,w z h ),h.

20 80 H. Duan et al. p p 0 + w w 0 p q h 0 + w z h 0 + p q h 0 + w z h 0 p q h 0 + w z h 0 + C (p q h,w z h ),h C (p q h,w z h ),h, which completes the proof. Corollary 4.3 For p and w as in Theorem 4.2 we have p p 0 + w w 0 Ch r ( p 1 + w 1 ), (4.63) p r + w r C ( p 1 + w 1 ). (4.64) If, additionally, p,w H 1+r ( ), we have p p 0 + w w 0 Ch 2r ( p 1+r + w 1+r ). (4.65) Proof For p,w H 1+s ( ), 0 s, there are p h Q h,w h W h such that [4,5,10, 15,26,39] K 1/2 p p h 2 0,K Ch s p 1+s, (4.66) K T h h 2 K T h h 2 K w w h 2 0,K 1/2 Ch s w 1+s. (4.67) p h p 0 Ch t p t, w h w 0 Ch t w t 0 t 1 + s, (4.68) p h t C p t, w h t C w t 0 t 1. (4.69) We then obtain (4.63) and (4.65) from (4.62), (4.66) and (4.67) with s = 0 and s = r, respectively, and (4.68) with t = 1 and t = 1 + r, respectively, where we used the fact that h h r and h 1+r h 2 r for r [0, 1] and 0 < h 1. Noting that the inverse estimates hold on finite dimensional spaces V h h r v h r C v h 0 v h V h, from (4.68) and (4.69) with t = r and (4.63), and p r C p 1,wehave p r p p h r + p h r C(h r p p h 0 + p r ) Ch r ( p p 0 + p p h 0 ) + C p r C ( p 1 + w 1 ), that is p r C( p 1 + w 1 ).

21 C 0 elements for generalized indefinite Maxwell equations 81 Similarly, w r C( p 1 + w 1 ). It then follows that (4.64) holds. Remark 4.5 We call p, w as the dual Fortin-type interpolations of p,w. The error estimates in Corollary 4.3 will play a key role in the establishment of the error estimates in the next section. 5 Coercivity and error estimates With the Fortin-type interpolations established in the previous section we are now in a position to investigate the coercivity property and to analyze the error estimates associated with problem (2.13). 5.1 Coercivity In this subsection we are to investigate the coercivity property of the cur/div part of the bilinear form in (2.14), that is, (R h (curl u), R h (curl v)) 0,h +( R h (div u), R h (div v)) 0,h. Proposition 5.1 [10,14,25,41] With those quadrature rules in Remark 2.1, letting the induced norm q 2 0,h := (q, q) 0,h q V h, (5.1) we have the following norm-equivalence and approximation properties: C 1 q 0 q 0,h C 2 q 0 q V h, (p, q) (p, q) 0,h Ch 2s p s q s p, q V h, 0 s 1. Theorem 5.1 We have R h (curl v h ) 2 0,h + R h (div v h ) 2 0,h C v h 2 0 v h K h (b), (5.2) where K h (b) is given in (4.30). Proof From Proposition 3.2 we write v h as the following L 2 orthogonal decomposition: v h = curl z v 0 v 0 H0 1 ( ), z H 1 ( )/R, v h 2 0 = v curl z 2 0, and let ṽ 0 Q h, z W h be the dual Fortin-type interpolations of v 0, z, respectively, i.e.

22 82 H. Duan et al. b(v h ; (ṽ 0, z)) b(v h ; (v 0, z)) = 0 v h K h (b), such that (see Corollary 4.3) ṽ 0 v z z 0 Ch r ( v z 1 ) Ch r v h 0, ṽ z 0 C ( v z 1 ) C v h 0. Let α>0beaconstant to be given. We have R h (curl v h ) 2 0,h = R h(curl v h ) α z 2 0,h + 2α(R h(curl v h ), z) 0,h α 2 z 2 0,h, where (R h (curl v h ), z) 0,h = (v h, curl z) = (curl v h, z), (curl v h, z) = (curl v h, z z) + (curl v h, z), (curl v h, z) = (v h, curlz) = curl z 2 0 and we have 2α(R h (curl v h ), z) 0,h = 2α(curlv h, z z) + 2α curl z 2 0. On the other hand, with the same α, wehave R h (div v h ) 2 0,h = R h (div v h ) αṽ 0 2 0,h + 2α( R h (div v h ), ṽ 0 ) 0,h α 2 ṽ 0 2 0,h, where ( R h (div v h ), ṽ 0 ) 0,h = (v h, ṽ 0 ) = (div v h, ṽ 0 ) = (div v h, ṽ 0 v 0 ) + (div v h,v 0 ), (div v h,v 0 ) = (v h, v 0 ) = v 0 2 0, and we have 2α( R h (div v h ), ṽ 0 ) 0,h = 2α(div v h, ṽ 0 v 0 ) + 2α v We therefore have R h (curl v h ) 2 0,h + R h (div v h ) 2 0,h = R h(curl v h ) α z 2 0,h + R h (div v h ) αṽ 0 2 0,h + 2α {(curl v h, z z) + (div v h, ṽ 0 v 0 )} +2α( v curl z 2 0 ) α2 ( ṽ 0 2 0,h + z 2 0,h ), where (curl v h, z z) + (div v h, ṽ 0 v 0 ) = b(v h ; (ṽ 0, z)) b(v h ; (v 0, z)) = 0, v curl z 2 0 = v h 2 0, ṽ 0 2 0,h + z 2 0,h C( ṽ z 2 0 ) C v h 2 0.

23 C 0 elements for generalized indefinite Maxwell equations 83 Hence, by choosing a suitable α>0, we have for all v h K h (b) R h (curl v h ) 2 0,h + R h (div v h ) 2 0,h α(2 αc) v h 2 0. The proof is finished. Remark 5.1 For the problem (1.1) with λ<0, we have from (5.2) that problem (2.13) has a unique solution. In the case λ = 0, a stabilization in [23] is needed. Regarding the indefinite case where λ>0, the existence and uniqueness follows from Theorem 5.2 below, see Corollary L 2 interpolation and regularity results In this subsection we study the L 2 projection with respect to the discrete L 2 inner product (, ) 0,h and the regularity results of problem (1.1) and (1.2). Proposition 5.2 Given p H s ( ), s 0. Let p h V h be defined by (p h, q) 0,h = (p, q) q V h, then p h 0,h C p 0, p p h 0 Ch s p s, (5.3) p h s C p s where 0 s 1. (5.4) Proof From the definition of p h and Proposition 5.1 we have p h 2 0,h = (p h, p h ) 0,h = (p, p h ) p 0 p h 0 C p 0 p 0,h, and we have p h 0,h C p 0. Let P h p V h be the L 2 projection of p, defined by (P h p, q) = (p, q) q V h. (5.5) We know from [40] that p P h p 0 Ch s p s, P h p s C p s where 0 s 1.

24 84 H. Duan et al. Then, for all q V h, from (5.5) and Proposition 5.1 we have (p h P h p, q) 0,h = (p, q) (P h p, q) 0,h = (p P h p, q) + (P h p, q) (P h p, q) 0,h p P h p 0 q 0 + Ch s P h p s q 0 Ch s p s q 0 Ch s p s q 0,h, from which by taking q = p h P h p we have and we have p h P h p 0,h Ch s p s, p h P h p 0 C p h P h p 0,h Ch s p s. Therefore, by triangle-inequality we have It then follows that p p h 0 p P h p 0 + P h p p h 0 Ch s p s. p h s p h P h p s + P h p s Ch s p h P h p 0 + C p s C p s. where we have used the inverse estimates on finite dimensional space V h h s v h s C v h 0 v h V h. Remark 5.2 For v H 0 (curl; ) H(div; ), we see that R h (curl v) W h, R h (div v) Q h are the L 2 projections which are similar to the one defined in Proposition 5.2, i.e. We can then have (R h (curl v), z h ) 0,h = (curl v, z h ) z h W h, ( R h (div v),q h ) 0,h = (div v,q h ) q h Q h. R h (curl v) 0,h + R h (div v) 0,h C ( curl v 0 + div v 0 ). (5.6) If, additionally, curl v, div v H s ( ) for s 0, then for 0 s 1 R h (curl v) s + R h (div v) s C ( curl v s + div v s ). (5.7)

25 C 0 elements for generalized indefinite Maxwell equations 85 Proposition 5.3 Let u H 0 (curl; ) H(div; ) be the exact solution of problem (1.1) and (1.2). Then, under Assumption A, u (H r ( )) 2 for some r, satisfying u r + curl u r + div u 1 + curl u 1 C f 0. Proof Since u H 0 (curl; ) H(div; ), from Assumption A we know that u (H r ( )) 2 for some r 0, satisfying It then follows from Proposition 3.3 that u r C u curl;div. u r C f 0. On the other hand, noting that from (1.1) and (1.2) wehave curl curl u div u 2 0 = curl curl u div u 2 0 = f + λ u 2 0, we have from Propositions 3.3 that curl u 1 = curl curl u 0 f + λ u 0 C f 0, div u 1 f + λ u 0 C f 0. Since curl u H 1 ( )/R, div u H0 1 ( ), from Poincaré Friedrichs inequalities we have and we have curl u 0 C curl u 1, div u 0 C div u 1, curl u 1 + div u 1 C f 0. Let u denote the solution of problem (1.1) and (1.2). We put p := div u H 1 0 ( ) w := curl u H 1 ( )/R. We see that if f H(div; ), then, p H0 1 ( ) satisfies p = λ div u + div f in, p = 0 on Ɣ. (5.8) Proposition 5.4 Let u H 0 (curl; ) H(div; ) be the exact solution of problem (1.1) and (1.2). Assume that f (H r ( )) 2 H(div; ). With p = div u and w = curl u, under Assumption A, we have p,w H 1+r ( ), and p 1+r + w 1+r C ( f r + f 0;div ).

26 86 H. Duan et al. Proof From Remark 3.1 and Proposition 3.3 we know that p H 1+r ( ), satisfying p 1+r C ( div u 0 + div f 0 ) C f 0;div. While w satisfies curl w = p + λ u + f, we then know that w H 1+r ( ), and we have from Proposition 5.3 that w 1 + w 1+r C f 0 + C ( p 1+r + u r + f r ) C ( f r + f 0;div ). 5.3 Error estimates In this subsection we analyze the error bounds between the exact solution and the finite element solution. This mainly consists of the consistency error estimates in Lemma 5.1 and the L 2 error estimates in Theorem 5.2. Lemma 5.1 Let u H 0 (curl; ) H(div; ) and u h be the exact solution of problem (1.1), (1.2) and the finite element problem (2.13), respectively. Let p Q h and w W h be the dual Fortin-type interpolations of p = div u H0 1 ( ) and w = curl u H 1 ( )/R, respectively. Then L h (u u h,v)= (w, R h (curlv)) + (p, R h (div v)) b(v; (p,w)) v U h, (5.9) where b is defined by (4.1).Forv K h (b) we further have L h (u u h,v) [ w w 0 + p p 0 + Ch r ( p 1 + w 1 ) ] (5.10) ( R h (curl v) 0,h + R h (div v) 0,h ), L h (u u h,v) ( w w 0 + p p 0 )( R h (curl v) 0,h + R h (div v) 0,h ) +C h 2r ( p 1 + w 1 )( R h (curl v) r + R h (div v) r ). (5.11) Proof For all v U h we have from (2.13), (1.1) and (1.2) that L h (u h,v) = ( f,v) = (curl u, curl v) + (div u, div v) λ(u,v) = b(v; (p,w)) λ(u,v). On the other hand L h (u,v) = (R h (curl u), R h (curl v)) 0,h + ( R h (div u), R h (div v)) 0,h λ(u,v) = (curl u, R h (curl v)) + (div u, R h (div v)) λ(u,v) = (w, R h (curl v)) + (p, R h (div v)) λ(u,v),

27 C 0 elements for generalized indefinite Maxwell equations 87 and we have L h (u u h,v)= (w, R h (curl v)) + (p, R h (div v)) b(v; (p, w)). Since p Q h, w W h are the dual Fortin-type interpolations of p and w, respectively, i.e., we have for all v K h (b) b(v; ( p, w)) = b(v; (p,w)) v K h (b), L h (u u h,v) = (w w, R h (curl v)) + (p p, R h (div v)) +( w, R h (curl v)) + ( p, R h (div v)) b(v; ( p, w)), = (w w, R h (curl v)) + (p p, R h (div v)) +( w, R h (curl v)) + ( p, R h (div v)) ( w, R h (curl v)) 0,h ( p, R h (div v)) 0,h, where, from Proposition 5.1 ( w, R h (curl v)) ( w, R h (curl v)) 0,h Ch r w r R h (curl v) 0 Ch r w r R h (curl v) 0,h, ( p, R h (div v)) ( p, R h (div v)) 0,h Ch r p r R h (div v) 0 Ch r p r R h (div v) 0,h, ( w, R h (curl v)) ( w, R h (curl v)) 0,h Ch 2r w r R h (curl v) r, ( p, R h (div v)) ( p, R h (div v)) 0,h Ch 2r p r R h (div v) r. We then have from Corollary 4.3 and Proposition 5.1 that L h (u u h,v) ( w w 0 + p p 0 )( R h (curl v) 0 + R h (div v) 0 ) +C h r w r R h (curl v) 0,h + Ch r p r R h (div v) 0,h [ w w 0 + p p 0 + Ch r ( p 1 + w 1 ) ] ( R h (curl v) 0,h + R h (div v) 0,h ), L h (u u h,v) ( w w 0 + p p 0 )( R h (curl v) 0 + R h (div v) 0 ) +C h 2r w r R h (curl v) r + Ch 2 r p r R h (div v) r ( w w 0 + p p 0 )( R h (curl v) 0,h + R h (div v) 0,h ) +C h 2r ( p 1 + w 1 )( R h (curl v) r + R h (div v) r ). Lemma 5.2 Let u H 0 (curl; ) H(div; ) and u h U h be the exact solution and the finite element solution. Let ũ U h be the Fortin-type interpolation of u. For v h = ũ u h = v 0,h + v 0,h with v 0,h K h (b) and v 0,h K h(b), under

28 88 H. Duan et al. Assumption A, we have L h (v h,v0,h ) Chr ( u curl;div + p 1 + w 1 ) ( R h (curl v0,h ) 0,h + R h (div v0,h ) 0,h). (5.12) Proof Under Assumption A, wehaveu (H r ( )) 2 and u r C u curl;div.we have L h (v h,v0,h ) = L h(ũ u h,v0,h ) = L h(ũ u,v0,h ) + L h(u u h,v0,h ). (5.13) Corollary 4.3 and Lemma 5.1 with v 0,h K h(b) lead to L h (u u h,v0,h ) C( p p 0 + w w 0 + h r ( p 1 + w 1 )) ( R h (curl v0,h )) 0,h + R h (div v0,h ) 0,h) (5.14) Ch r ( p 1 + w 1 )( R h (curl v0,h )) 0,h + R h (div v0,h ) 0,h), while from Remark 4.1 and Theorem 4.1 and the coercivity in Theorem 5.1,wehave L h (ũ u,v 0,h ) = λ(ũ u,v 0,h ) Chr u curl;div v 0,h 0 Ch r u curl;div ( R h (curl v 0,h ) 0,h + R h (div v 0,h ) 0,h). Hence, combining (5.13) (5.15) to have the desired (5.12). (5.15) Let u h U h be the finite element solution of (2.13), and ũ U h be the Fortin-type interpolation of the exact solution u H 0 (curl; ) H(div; ) of problem (1.1) and (1.2). Writing ũ = ũ 0,h + ũ 0,h with ũ 0,h K h (b), ũ 0,h K h(b) and u h = u 0,h + u 0,h with u 0,h K h (b), u 0,h K h(b), we introduce v h := ũ u h = v 0,h + v 0,h, (5.16) with v 0,h = ũ 0,h u 0,h K h (b), v 0,h = ũ 0,h u 0,h K h(b). For the duality argument, choosing the v 0,h K h(b), we consider the following auxiliary problem: To find u H 0 (curl; ) H(div; ) such that curl curl u div u λ u = v0,h in, div u = 0, u τ = 0 onɣ. (5.17) Such choice of v 0,h is different from the usual choice of u u h and is not even the choice of v h itself. For the latter two choices we have not been able to obtain the L 2 error estimates in Theorem 5.2 below. It suffices to establish the estimates about v 0,h, though. Let p := div u H 1 0 ( ), w := curl u H 1 ( )/R, (5.18)

29 C 0 elements for generalized indefinite Maxwell equations 89 we have from Propositions 5.3 and 3.3 u r + curl u r + div u r + u curl;div + p 1 + w 1 C v0,h 0. (5.19) Theorem 5.2 Let v0,h be defined as in (5.16). Under Assumption A, for all h < h with h < 1 sufficiently small we have v0,h 0 Ch 2r ( u curl;div + p 1 + w 1 ) + C( w w 0 + p p 0 ), (5.20) where p = div u H 1 0 ( ), w = curl u H 1 ( )/R, and p Q h, w W h are the dual Fortin-type interpolations of p,w, respectively, while u H 0 (curl; ) H(div; ) (H r ( )) 2 is the exact solution. Moreover, we have R h (curl (u u h )) 0,h + R h (div (u u h )) 0,h = R h (curl (u u 0,h )) 0,h + R h (div (u u 0,h )) 0,h Ch r ( u curl;div + p 1 + w 1 ), (5.21) where u h = u 0,h + u 0,h is the finite element solution, with u 0,h K h (b), u 0,h K h (b). Proof From (5.17) wehave v0,h 2 0 = (curl u, curl v0,h ) + (div u, div v0,h ) λ(u,v0,h ) = b(v0,h ; (p,w )) λ(u,v0,h ) = (R h (curl u ), R h (curl v0,h )) 0,h + ( R h (div u ), R h (div v0,h )) 0,h (5.22) λ(u,v0,h ) + b(v 0,h ; (p,w )) (R h (curl u ), R h (curl v0,h )) 0,h where ( R h (div u ), R h (div v 0,h )) 0,h := I 1 + I 2, I 1 = (R h (curl u ), R h (curl v0,h )) 0,h + ( R h (div u ), R h (div v0,h )) 0,h λ(u,v0,h ), I 2 = b(v0,h ; (p,w )) (R h (curl u ), R h (curl v0,h )) 0,h ( R h (div u ), R h (div v0,h )) 0,h. In the subsequent we estimate I 1 and I 2. The estimates are divided into two steps. Step 1. To estimate I 1. Let u h U h be the Fortin-type interpolation of u, and write it as the following decomposition: u h = u 0,h + u 0,h, (5.23)

30 90 H. Duan et al. where u 0,h K h(b), u 0,h K h (b), satisfying (see Remarks 4.1, 4.2 and 4.4, Theorem 4.1) We have R h (curl (u u h )) = R h(curl (u u 0,h )) = 0, R h (div(u u h )) = R h (div(u u 0,h )) = 0, (5.24) u u h 0 Ch r u curl;div. (5.25) I 1 = (R h (curl u ), R h (curl v0,h )) 0,h + ( R h (div u ), R h (div v0,h )) 0,h λ(u,v0,h ) = (R h (curl u h ), R h(curl v0,h )) 0,h + ( R h (div u h ), R h (div v0,h )) 0,h λ(u h,v 0,h ) λ(u u h,v 0,h ) = (R h (curl u 0,h ), R h (curl v h )) 0,h + ( R h (div u 0,h ), R h (div v h )) 0,h λ(u 0,h,v h ) λ(u u h,v 0,h ) (5.26) = (R h (curl u 0,h ), R h (curl (u u h ))) 0,h + ( R h (div u 0,h ), R h (div (u u h ))) 0,h λ(u 0,h, u u h ) λ(u 0,h, ũ u) λ(u u h,v 0,h ) = L h (u u h, u 0,h ) λ(u 0,h, ũ u) λ(u u h,v 0,h ), wherewehaveusedv h = ũ u h = ũ u + u u h and the fact from Remarks 4.1, 4.2 and 4.4 that the Fortin-type interpolation ũ = ũ 0,h + ũ 0,h with ũ 0,h K h (b), ũ 0,h K h(b) (5.27) of the exact solution u satisfies R h (curl(u ũ)) = R h (curl(u ũ 0,h )) = 0, R h (div(u ũ)) = R h (div(u ũ 0,h )) = 0. (5.28) Define u 0,h the element-local L 2 projection of u 0,h by u 0,h K = K u 0,h K K T h, (5.29) for which we have the interpolation property u 0,h u 0,h 0 Ch r u 0,h r. (5.30) But, since u h = u 0,h +u 0,h, with u 0,h K h(b), u 0,h K h (b), is the Fortin-type interpolation of u, we have from Remark 4.4 that u 0,h r C u curl;div.

31 C 0 elements for generalized indefinite Maxwell equations 91 Hence u 0,h u 0,h 0 Ch r u 0,h r Ch r u curl;div Ch r v0,h 0, (5.31) and we have λ(u 0,h, ũ u) = λ(u 0,h u 0,h, ũ u) Ch 2r v0,h 0 u curl;div, (5.32) wherewehaveused(4.5) (or(4.6) for quadrilaterals) and (4.7) asfollows: (ũ u) = 0 K T h, u ũ 0 Ch r u r C u curl;div. K We also have from (5.25) and (5.19) λ(u u h,v 0,h ) Chr u curl;div v0,h 0 Ch r v0,h 2 0. (5.33) On the other hand, from Lemma 5.1 we have, for u 0,h K h (b), L h (u u h, u 0,h ) ( w w 0 + p p 0 )( R h (curl u 0,h ) 0,h + R h (div u 0,h ) 0,h ) + Ch 2r ( p 1 + w 1 )( R h (curl u 0,h ) r + R h (div u 0,h ) r ), where, from Remark 5.2 and (5.24) and (5.19), R h (curl u 0,h ) 0,h + R h (div u 0,h ) 0,h = R h (curl u h ) 0,h + R h (div u h ) 0,h = R h (curl u ) 0,h + R h (div u ) 0,h C( curl u 0 + div u 0 ) C v 0,h 0. R h (curl u 0,h ) r + R h (div u 0,h ) r = R h (curl u h ) r + R h (div u h ) r = R h (curl u ) r + R h (div u ) r C( curl u r + div u r ) C v 0,h 0. We thus have L h (u u h, u 0,h ) Ch 2r ( p 1 + w 1 ) v 0,h 0 + C ( p p 0 + w w 0 ) v 0,h 0. (5.34) Therefore, from (5.22), (5.32), (5.33) and (5.34) to get I 1 Ch r v 0,h Ch2r v 0,h 0 u curl;div + Ch 2r ( p 1 + w 1 ) v 0,h 0 +C ( p p 0 + w w 0 ) v 0,h 0. (5.35)