Analysis of A Continuous inite Element Metod for Hcurl, div)-elliptic Interface Problem Huoyuan Duan, Ping Lin, and Roger C. E. Tan Abstract In tis paper, we develop a continuous finite element metod for te curlcurl-grad div vector second-order elliptic problem in a tree-dimensional polyedral domain occupied by discontinuous nonomogeneous anisotropic materials. In spite of te fact tat te curlcurl-grad div interface problem is closely related to te elliptic interface problem of vector Laplace operator type, te continuous finite element discretization of te standard variational problem of te former generally fails to give a correct solution even in te case of omogeneous media wenever te pysical domain as reentrant corners and edges. To discretize te curlcurl-grad div interface problem by te continuous finite element metod, we apply an element-local L projector to te curl operator and a pseudo-local L projector to te div operator, were te continuous Lagrange linear element enriced by suitable element and face bubbles may be employed. It is sown tat te finite element problem retains te same coercivity property as te continuous problem. An error estimate O r ) in an energy norm is obtained between te analytical solution and te continuous finite element solution, were te analytical solution is in L Hr Ω l )) 3 for some r 1/, 1] due to te domain boundary reentrant corners and edges e.g., nonconvex polyedron) and due to te interfaces between te different material domains in Ω = L Ω l. 1 Introduction In tis paper we sall develop a continuous finite element metod for te curlcurl-grad div interface problem in te pysical domain Ω R 3. To model different anisotropic nonomogeneous materials filling Ω, we introduce two coefficient matrices µ, ε R 3 3, describing te pysical properties, e.g., permeability and permittivity of te materials in electromagnetism. Te curlcurl-grad div second-order elliptic system tat we sall consider reads as follows: curl µ 1 curl u ε div ε u = f in Ω, 1.1) were u represents te unknown e.g., electric field in electromagnetism), f L Ω)) 3 denotes te known source function, te curl operator curl v = v, and te div operator div v = v, wit denoting te gradient operator. Let ε and µ be piecewise smoot functions, i.e., tere exists a partition of Ω into a finite number of subdomains Ω l, 1 l L, were µ and ε are smoot in all material subdomains Ω l, but tey are possibly discontinuous and differ greatly in magnitude across inter-subdomain-boundaries. We sall assume tat Ω and Ω l are simply-connected tree-dimensional Lipscitz polyedra wit connected boundaries. As usual, it is assumed tat µ, ε are symmetric, uniform boundedness, and piecewise smoot, and satisfy uniform ellipticity property, i.e., letting ω = ω ij ) R 3 3 denote any of µ and ε, we ave ω ij = ω ji, ω ij L Ω), ω ij W 1, Ω l ), 1 l L, 1 i, j 3, Scool of Matematical Sciences, Nankai University, Tianjin 300071, Cina. E-mail: yduan@nankai.edu.cn Department of Matematics, University of Dundee, Dundee DD1 4HN, UK. E-mail:plin@mats.dundee.ac.uk Department of Matematics, National University of Singapore, Science Drive, Singapore 117543. E-mail: scitance@nus.edu.sg 1
and tere exists a constant C ω > 0 suc tat C 1 ω ξ ξ T ωx)ξ C ω ξ, a.e. in Ω, ξ R 3, were, from te above, te inverse ω 1 of ω satisfies te same: Cω 1 ξ ξ T ω 1 x)ξ C ω ξ a.e. in Ω and for all ξ R 3, and L p D) and W s,p D) are classical Sobolev spaces over an open subset D Ω, see [1]. Denote by Γ te boundary of Ω and by Γ ij = Ω i Ω j te material interfaces. Let n denote te outward unit normal vector to Γ or te unit normal vector to Γ ij oriented from Ω i to Ω j. Along te interface Γ ij, we define jumps [v n] Γij = v n Ωi v n Ωj, [v n] Γij = v n Ωi v n Ωj, and [q] Γij = q Ωi q Ωj. We supplement equation 1.1) wit te following boundary conditions u n = 0, div ε u = 0 on Γ. 1.) Te unknown u additionally satisfies te following pysical jump conditions on te interfaces [u n] = 0, [µ 1 curl u n] = 0, [ε u n] = 0, [div ε u] = 0 on Γ ij. 1.3) We see tat u belongs to H 0 curl; Ω) Hdiv; ε; Ω), wit te curl and div Hilbert spaces defined as in [39], Hcurl; Ω) = {v L Ω)) 3 : curl v L Ω)) 3 }, H 0 curl; Ω) = {v Hcurl; Ω) : v n Γ = 0}, Hdiv; ε; Ω) = {v L Ω)) 3 : div ε v L Ω)}. Equation 1.1) is quite general, and it covers a wide range of models and forms in many pysical problems involving curl and div operators. Two typical examples of 1.1) are from electromagnetism and fluid-structure interaction problems. In te case of vector potential Maxwell s equations, a model reads See [1, 0]): curl µ 1 curl u = f, div ε u = 0 in Ω. 1.4) A very detailed development and explanations of te problem 1.1)-1.3) for time-armonic Maxwell s equations in eterogeneous materials can be found in Section 1 of [31] and Section 7 of [9]. In fact, te strong form of te variational formulation in [31] and [9] is exactly te same as 1.1)-1.3), except tat ere te electromagnetic wave number/frequency is taken as zero. Anoter example wic may be stated as te form 1.1) is te simplest model for fluid-structure interaction problem as followssee [54]): div u = f, curl u = 0. 1.5) At te same time, equation 1.1) is widely-used for teoretical studies of many matematical issues relating to te Maxwell s equations, including regularity-singularities See [9, 11, 5, 30, 31]), solvability-uniqueness See [1, 9, 41, 7,, 0, 36, 59, 60, 19, 53]), and numerical metods See [8, 4, 15, 45]). Terefore, te numerical metod of 1.1) is a very important subject and may be used to motivate and justify tose teoretical studies. Also, equation 1.1) is closely related to te second-order elliptic problem of vector Laplace operator. As a matter of fact, if µ = ε = 1, using te relationsip were is te Laplace operator, we obtain a vector Poisson equation u = curl curl u div u, 1.6) u = f. 1.7) Additionally, if Ω is smoot enoug or is convex polyedron, ten te solution of problem 1.1) and 1.) wit µ = ε = 1 is te same as tat of te vector Poisson problem 1.7) and 1.), and, moreover, te solution is in H 1 Ω)) 3 See [9] for more details). inite element discretizations of all tese equations are not so simple as tey look, toug. As an illustration, let us consider te omogeneous case of 1.1) were µ = ε = 1. Note tat te analytical solution
u belongs to Hcurl; Ω) Hdiv; Ω), were Hdiv; Ω) = Hdiv; 1; Ω). Tus, a natural coice of finite elements is te continuous Lagrange element as in te book [4], because a Hcurl; Ω) Hdiv; Ω) conforming piecewise-polynomial element is necessarily in H 1 Ω)) 3 and continuous. As is well known, indeed, it as been matematically and practically justified in te literature [4, 47, 63, 48, 49] tat te continuous finite element metod is te most popular numerical metod for solving 1.7), wic is equivalent to 1.1) in a smoot domain or a convex polyedron. Unfortunately, wenever te domain as reentrant corners/edges/ interface corners, te solution u is usually not in H 1 Ω)) 3, instead, it is in a muc larger H r Ω)) 3 for some r < 1, see [9, 30, 31, 11, 5, 9]. It is tis very low regularity tat causes enormous difficulties in te finite element solution of 1.1). By te continuous finite element metod an incorrect solution may be obtained te continuous finite element solution converges to a function of H 1 Ω)) 3, but not to te solution wic does not belong to H 1 Ω)) 3, see [30, 41, 11, 43, 4]. It is better understood now tat te reason is mainly at te standard variational formulation of 1.1) [49, 36, 41, 33, 9, 31, 9, 5]: µ 1 curl u, curlv) + div ε u, div ε v) = f, v), 1.8) were u, v H 0 curl; Ω) Hdiv; ε; Ω) and, ) denotes te L inner product. Altoug tis incorrect convergence as been known for quite some time in bot matematical and engineering communities, attempts ave never been stopped to andle te low regularity solution wit continuous elements, because tere are so many well-known popular advantages of te continuous Lagrange elements. However, only over te last decade, tree different continuous finite element metods ave been developed wic can give correct convergent finite element approximations of te low regularity solution. Te first metod of reabilitation of nodal finite element is proposed in [8], i.e., te weigted regularization metod for Maxwell s problem 1.4) wit µ = ε = 1, were a weigt function is applied to te div term. It uses a continuous finite element space including te gradient of some C 1 element [4]. Similar weigted metods ave since been developed in [16, 50]. Te second metod is te H 1 least-squares metod [13] for te first-order system wit a given f and g 0 curl u = f, div ε u = g 0, 1.9) We note tat 1.9) may be also formulated into a least-squares variational problem according to te L inner products, resulting in te same bilinear form as 1.8), wit µ = 1. In tis least-squares formulation, te rigt-and side of 1.8) sould be accordingly replaced by f, curlv) + g 0, divεv). Te H 1 metod is to reformulate 1.9) into a least-squares variational problem based on te so-called minus-one inner products, instead of te L inner products. Te tird metod is te element-local L projection metod in [33] for te Maxwell s problem 1.4) wit µ = ε = 1, were two element-local L projectors are applied to bot curl and div operators. In most real applications, owever, problem 1.1) is imposed in discontinuous, nonomogeneous, anisotropic media. Most of te aforementioned continuous finite element metods eiter are not directly applicable or become inefficient for te general problem 1.1), due to different pysical caracteristics reflected by µ and ε of different media. In fact, te standard variational formulation 1.8) itself is not suitable for direct continuous finite element discretizations in te case of discontinuous ε, because div ε v is not generally welldefined for continuous piecewise-polynomials v. We ave so far not seen ow to apply te H 1 least-squares metod of te first-order system 1.9) to te general curlcurl-grad div second-order system 1.1) even if µ = ε = 1. Wen te element local L projection metod is directly applied to te general problem 1.1) wit discontinuous ε, a discontinuous element metod prevails [34]. Tis migt be not consistent wit te original intention of te use of te continuous elements. Tere are a lot of finite element metods for scalar second-order elliptic interface problems see, e.g., [18, 51]) of Laplace operator type. But, te scalar solution terein at least belongs to H 1 Ω), and tose finite element metods cannot be employed for problem 1.1). Tis is not only because te solution of 1.1) is often in H r Ω)) 3 for some r < 1, but also because 1.1) is in general not a problem of vector Laplace operator type unless µ = ε = 1, cf. 1.6), 1.7)). It is also very interesting to mention a few non or nearly C 0 finite element metods for te Maxwell s interface problems in te case of piecewise constant coefficients. An edge element mixed/indefinite metod is proposed in [] for problem 1.4), were te classical edge or Nédélec 3
element in [55, 56] is used. In [3], a finite volume mixed/indefinite metod is analyzed for te first-order systems of problem 1.4) in terms of electric and magnetic fields. A Singular unction metod in [8] is considered for problem 1.9). See also an alternate Singular unction metod in [4]. A nearly continuous mixed finite element metod in te context of te weigted regularization metod is also recently presented in [5] for te eterogeneous time-armonic Maxwell s equations. In tis paper, we sall develop a continuous finite element metod for te general problem 1.1) wit discontinuous µ and ε. We employ te continuous Lagrange linear element in [4] enriced wit element and face bubbles in order to approximate te low regularity solution. Wit two L projections applied to bot curl and div operators, plus a stabilization term S u, v), we propose te following finite element formulation R µ 1 curl u), R µ 1 curl v)) µ + R div ε u), R div ε v)) 0, + S u, v) = f, v). 1.10) Te operator R for te curl operator is an element-local L projection wic is defined element by element onto a discontinuous linear finite element space wit respect to te µ-weigted L inner product, ) µ, wile R for te div operator is a pseudo-local L projection wic is defined onto a continuous linear finite element space, wit respect to te mass-lumping L inner product, ) 0, te resulting matrix is ten diagonal). Te stabilization term S is defined as te element-wise analogue of te bilinear form of 1.8) weigted wit local mes sizes. Note tat te computational cost of 1.10) is not muc more tan tat of 1.8) given tat te latter can be directly discretized by te continuous finite element metod), since all te additions in 1.10), compared wit 1.8), are locally evaluated. We ave noticed tat te mes-locally-weigted stabilization is also used in a recent metod of H 1 type in [10]. Wit te stabilization role of S, we sow tat te same coercivity property as te continuous problem 1.8) olds for 1.10) a nice feature for using iterative metods to solve te resulting algebraic system. Just like te Poisson equation for Laplace operator, te associated condition number is O ), wit being te mes size. Bot L projectors R and R, togeter wit te element and face bubbles in te continuous finite element space of te solution, essentially remove te effects of curl and div partial derivatives on te analytical solutionsee Teorem 4.). Wit te Crouzeix-Raviart nonconforming linear element interpolations [3], an ad oc strategy is designed to estimate te errors associated wit te stabilization term wic is explicitly involved wit curl and div partial derivatives on te low regularity solution See Lemma 4.5). Terefore, wit respect to te regularity of te solution in material subdomains, i.e., u L Hr Ω l )) 3 for some 1/ < r 1, for a suitable regular rigt-and side f see a furter explanation below), we prove te optimal error estimates O r ) in an energy norm between te analytical solution of 1.1) and te continuous finite element solution of 1.10). We sould point out tat te two new key ingredients in 1.10) are te new stabilization S te elementwise analogue weigted wit local mes sizes) of te bilinear form of 1.8) and te new pseudo-local L projection R. It is because of tem tat not only can we ave a simple finite element metod, but also can we employ te continuous elements even if ε is discontinuous. To te autors knowledge, te continuous finite element metod in tis paper is te first attempt so far for te general curlcurl-grad div interface problem 1.1)-1.3) wit bot µ and ε being discontinuous and nonomogeneous and anisotropic. Before ending tis section we would like to make a remark about te regularity. Te regularity of te solution may eavily depend on tat of te coefficients, te domain and te rigt-and side. A general dependence of te regularity of te solution on µ 1, ε, Ω and f in te interface problem 1.1)-1.3) does not seem to be available in te literature, but readers may refer to [31] for some results in te case of te eterogeneous time-armonic Maxwell s equations. Since tis issue is beyond te scope of tis paper, we ave made two ypoteses Hypotesis H1) and Hypotesis H) on te regularity of te solution for tecnical needs wen deriving te error estimates. or example, we sall use a regular-singular decomposition, wic relies on te regularity of te domain boundary Γ, te material interfaces, and even te number of material subdomains, see [31]. In tis paper, te regularity assumptions for te solution, te coefficients and te rigt-and side can be stated as follows: u H 0 curl; Ω) Hdiv; ε; Ω) L Hr Ω l )) 3, µ 1 curlu Hcurl; Ω) H 0 div; µ; Ω) L Hr Ω l )) 3, ε, µ L Ω)) 3 3 L W 1, Ω l )) 3 3, and f L Ω)) 3 L Hdiv; Ω l) Hcurl; ε 1 ; Ω l ) H r Ω l )) 3, were Hcurl; ε 1 ; Ω l ) = {v L Ω l )) 3 : curl ε 1 v L Ω l )) 3 }. It is assumed tat te regularity index r > 1/. or te solution, tis is because te construction 4
of te finite element interpolation involves te element-face integrals of u wic are well-defined if T r > 1/. or te rigt-and side, tis is because te traces of f along te domain boundary and te material interfaces are used. We will see tat te additional regularity assumption on f, i.e., f L Hdiv; Ω l) Hcurl; ε 1 ; Ω l ) H r Ω l )) 3 can ensure tat divεu and µ 1 curlu ave suitable regularity, so tat we can derive te desired error estimates O r ) wen dealing wit te consistency errors caused by te L projections R and R, see subsection 4.1. On te oter and, as we ave noted, te variational formulation 1.8) of te curlcurl-grad div problem may also come from a first-order system curlu = f and divεu = g 0, like 1.9), wit µ = 1. In tat case, te rigt-and side of 1.10) becomes f, R curlv)) + g 0, R divεv)), up to an additional consistent rigt-and side corresponding to te stabilization term S see Remark.9), te just mentioned consistency errors from te L projections R and R do not exist. Tus, it suffices to assume te regularity as follows: u H 0 curl; Ω) Hdiv; ε; Ω) L Hr Ω l )) 3 for some r > 1/ and ε L Ω)) 3 3 L W 1, Ω l )) 3 3 and f L Ω)) 3 and g 0 L Ω). In oter words, for first-order systems, we do not require any additional regularity of curlu and any additional regularity on f and g 0. Te rest of te paper is organized as follows. In section, we define te L projected continuous finite element metod. Section 3 is devoted to te establisment of te coercivity property and te condition number of te resulting linear system. In section 4 we sow te error bounds of te proposed finite element metod in an energy norm. Some concluding remarks are given in te last section. Te finite element metod In tis section, we sall describe te finite element metod 1.10) for problem 1.1), 1.), 1.3)..1 inite element space We first define te continuous finite element space U. Let C denote a sape-regular conforming also conforming wit te material interfaces) triangulation [4] of Ω into tetraedra, wit diameters K for all K C bounded by. Let E be te collection of all element faces, and E Γ inter te collection of all te element faces on Γ, and E te collection of all te element faces on te discontinuous interfaces of te material ε. Denote E 0 = E \ E Γ E inter. or ease of exposition, we sall assume tat ε is a piecewise polynomial wit respect to te triangulation of C. In Section 4 we remark ow to deal wit a general piecewise smoot ε by its finite element interpolation. Let λ i, 1 i 4, denote te local basis of linear polynomial associated wit te it vertex of K, see [4]. Let denote a face of te element boundary K. If E Γ, we define b as te usual face bubble on, satisfying b H 1 K), b H0 1 ) on K, b = 0 on K \..1) or example, let a 1, a, a 3 be te tree vertices of te face, ten b = λ 1 λ λ 3. If E Γ, we put b = 0. or any given K, let b K denote te usual element bubble on K, i.e., b K H 1 0 K), for example, b K = λ 1 λ λ 3 λ 4. As usual, let P n be te space of polynomials of degree not greater tan n 0, see [4]. Let P 0 ε; K) := span{ε P 0 K)) 3, P 0 K)) 3 }..) Let H 1 Ω), H 1 0 Ω) denote te standard Hilbert spaces [1]. We introduce te following face-bubble space Φ and element-bubble space Θ : Φ := { v H 1 0 Ω)) 3 ; v K P 1 K)) 3 b 1, b, b 3, b 4 ), K C }.3) and Θ := {v H 1 0 Ω)) 3 ; v K P 0 ε; K) b K, K C }..4) 5
Let U := { v H 1 Ω)) 3 ; v K P 1 K)) 3, K C }..5) We define te continuous finite element space for approximating te solution as follows: U := U + Φ + Θ..6) Remark.1 Te role of te element and face bubbles elp to remove te effects of te curl and div partial derivatives on te solution See Section 4), so tat we can obtain a correct continuous finite element solution wen te analytical solution as a low regularity. Note tat tere is no omogeneous tangential boundary conditions imposed in U. Oterwise, te coexistence of several tangential vectors at te corners and edges along Γ makes it difficult to evaluate te tangential component of u U at tose places. Remark. We see tat for any v Φ on any given face, v P 1 )) 3 b. or any face E Γ, we ave v = 0, since b as been defined as zero in advance for boundary faces. In addition, any function in Θ is globally continuous altoug ε is discontinuous, because te element bubble is zero on any face including te material interfaces). Tus, even if ε is discontinuous, te finite element space U is continuous in te entire domain.. L projection In tis subsection, we sall define te L projections. Let L Ω) denote te Hilbert space of square integrable Lebesgue measurable functions, equipped wit L -inner product u, v) = Ω u v and L norm v 0 = v, v), see [1]. Define te finite element space of discontinuous piecewise polynomials of degree not greater tan n as follows: P n := {q L Ω); q K P n K), K C }..7) Let, ) 0, denote te mass-lumping L inner product on P 1 H1 0 Ω), i.e. p, q) 0, = K 4 4 pa i )qa i ),.8) i=1 were a i are te four vertices of K, and K is te volume of K. Let, ) µ denote te µ-weigted L inner product, i.e. u, v) µ = µ u v. Given v Hcurl; Ω), we define R div ε v) P 1 H1 0 Ω) and R µ 1 curlv) P 1 )3 as follows: Ω R div ε v), q) 0, := ε v, q) q P 1 H0 1 Ω),.9) R µ 1 curl v), q) µ := curl v, q) v n q, q P 1)3..10) Remark.3 R is element-locally defined. R is essentially local, called as pseudo-local, since te mass-lumping inner product.8) results in a diagonal matrix for basis functions in P 1 H1 0 Ω). Te approximation of te L inner product, ) by.8) is referred to as mass-lumping [6]. Remark.4 Let, ) 0,K denote te L inner product on K. By Green s formula, we ave from.9) and.10) R div ε v), q) 0, = div ε v, q) 0,K [εv n] q,.11) K C 6 E Γ
R µ 1 curl v), q) µ = v, curl q) 0,K + E 0 Einter v n [q]..1) rom.9) and.1) we see tat te curl and div partial derivatives no longer act on te solution. Tis is an essential strategy in dealing wit te low regularity solution. We also see tat R is well-defined even if ε is discontinuous. Tis is te key factor for wic we can use te continuous finite element space in te case were ε is discontinuous..3 Stabilization term In tis subsection, we sall define te stabilization term S. We define S u, v) = K div ε u, div ε v) 0,K + K curl u, curl v) 0,K + [ε u n], [ε v n]) 0, + u n, v n) 0,, E Γ.13) were, ) 0, denotes te L inner product on, and is te diameter of. Remark.5 We see tat.13) is te elementwise analogue of te bilinear form of 1.8) weigted wit local mes sizes. Te stabilization role of S is for te coercivity. Suc natural and simple S greatly simplifies te design of te L projection finite element metod but also reduces te complexity in te corresponding coercivity analysis. Consequently, te implementation of te underlying finite element metod is easier. Altoug te curl and div partial derivatives still exist in S, we can control tem reasonably well for te low regularity solution tat does not belong to H 1 Ω)) 3 See Section 4). Remark.6 or te analytical solution u H 0 curl; Ω) Hdiv; ε; Ω) of problem 1.1)-1.3), all te jumps, boundary terms in.10)-.13) disappear..4 inite element problem We are now in a position to state te finite element problem: find u U suc tat for all v U R µ 1 curl u ), R µ 1 curl v )) µ + R div ε u ), R div ε v )) 0, + S u, v ) = f, v )..14) Remark.7 We sould remark tat all te additions in.14) relative to te standard formulation 1.8) are locally evaluated. So, te realization of.14) would not incur muc more computational cost tan tat of 1.8). Remark.8 Denote by L te bilinear form of.14). We see tat it is symmetric and non-negative. Remark.9 Te finite element metod stated as above is also applied to first-order systems, suc as 1.9) were µ = 1. In tat case, as mentioned in te Introduction section, te rigt-and side in.14) sould be correspondingly replaced by te following: f, R curlv )) + g 0, R divεv )) + R f, g 0 ; v ), were R f, g 0 ; v ) := K g 0, div ε v) 0,K + K f, curl v) 0,K. or te first-order system 1.9), problem.14) is consistent in te usual sense [4]. In general, problem.14) incurs consistency errors from te L projections R and R and te stabilization term S. or te rest of tis paper, we sall only analyze problem.14), including te coercivity, te condition number, te consistency error, te construction of te finite element interpolation, and te error bound. All te analysis is completely te same wen applied to te first-order system, except tat te consistency error does not exist. We also point out tat it is te consistency error from te L projections tat requires more regularity on f tan L Ω)) 3, see subsection 4.1. See also Remark 4.6. Remark.10 We sould note tat if te continuous finite element space of te solution is cosen to include te gradient of some C 1 element, ten te L projection R is unnecessary and can be dropped, and, meanwile, te sub-term associated wit te curl operator in te stabilization term S can also be dropped. 7
3 Coercivity and condition number In tis section, we sall investigate te coercivity and te condition number of.14). 3.1 Coercivity Tis subsection is devoted to te coercivity property of.14). Introduce div Hilbert spaces [39]: H 0 div; Ω) = {v Hdiv; Ω) : v n Γ = 0}, H 0 div 0 ; Ω) = {v H 0 div; Ω) : div v = 0}, and te ε-weigted L inner product, ) ε : u, v) ε = Proposition 3.1 See [9, 11, 31]). or any ψ Hcurl; Ω) H 0 div; Ω), it can be written into te following regular-singular decomposition: Ω ε u v. ψ = ψ 0 + p 0, ψ 0 H 1 Ω)) 3, p 0 H 1 Ω)/R, were ψ 0 satisfies ψ 0 1 C { curl ψ 0 + div ψ 0 }. Proposition 3. See [35, 3, 59, 60]) or any v L Ω)) 3, we ave te following L ortogonal decomposition wit respect to te ε-weigted L inner product, ) ε : v = p + ε 1 curl ψ, were p H 1 0 Ω), ψ Hcurl; Ω) H 0 div 0 ; Ω), satisfying ε 1 v 0 = ε 1 p 0 + ε 1 curl ψ 0, ψ 0 C curl ψ 0. Proposition 3.3 See [6]) or te L norm 0, induced from te mass-lumping L inner product.8), we ave te following norm-equivalence property C q 0, q 0 C 1 q 0, q P 1 H 1 0 Ω). Teorem 3.1 We ave As a consequence, problem.14) as a unique solution in U. Proof. Noting tat L v, v) C v 0 v U. 3.1) L v, v) = R div εv) 0, + R µ 1 curl v) µ + S v, v), were µ is te induced norm of te µ-weigted L inner product, ) µ, we need only to sow R div ε v) 0, + R µ 1 curl v) µ C 1 v 0 C S v, v) v U. 3.) In fact, if 3.) olds true, from 3.) we can obtain L v, v) 1 max1, C ) R ) div εv) 0, + R µ 1 curl v) C 1 µ + C S v, v) max1, C ) v 0. 3.3) 8
To sow 3.), from Proposition 3. we write v as te following L -ortogonal decomposition wit respect to te ε-weigted L inner product, ) ε : wit p H 1 0 Ω), ψ Hcurl; Ω) H 0 div 0 ; Ω), satisfying and rom Proposition 3.1 we furter write ψ as were ψ 0 H 1 Ω)) 3, curl ψ 1 = 0, and we ave v = p + ε 1 curl ψ, 3.4) ε 1 v 0 = ε 1 p 0 + ε 1 curl ψ 0 3.5) ψ 0 C curl ψ 0. 3.6) ψ = ψ 0 + ψ 1, 3.7) ψ 0 1 C curl ψ 0. 3.8) According to te two components p, ψ) in te decomposition of v, we divide te proof into two steps. Te first step on p and te second step on ψ give lower bounds for R div ε v) 0, and R µ 1 curl v) µ, respectively. Step 1. We consider p. We take p P 1 H1 0 Ω) te interpolation of p, suc tat See [7, 6, 61]) ) 1 K p p 0,K + 1 p p 0, C 3 p 1, 3.9) E p 0 C 3 p 1. 3.10) Note tat, to construct te above p for p H0 1 Ω), we may employ te Clément interpolation in [6], or Bernardi-Girault interpolation [7], or Scott-Zang interpolation [61]. or carrying out te analysis, we recall tat te Poincaré-riedrics inequality p 1 C P p 0 olds for all p H 1 0 Ω) wit te constant C P and tat te uniform ellipticity of ε: v 0 C 1/ ε for all v L Ω)) 3 wit te constant C ε as indicated in te Introduction section. Let δ > 0 be a constant to be determined. We ave ε 1/ v 0 olds R div ε v) 0, = R div ε v) + δ p 0, δ p 0, δ R div ε v), p) 0,. 3.11) rom p 0, C 1 p 0 in Proposition 3.3), and 3.10), we ave p 0, C 1 p 0 C 1 C 3 p 1 C 1 C 3 C P p 0 C 1 C 3 C P C 1/ ε ε 1 p 0, i.e., putting C 4 := C 1 C 3 C P Cε 1/, we ave p 0, C 4 ε 1 p 0. 3.1) We ave δ R div ε v), p) 0, = δ ε v, p) = δ div ε v, p) 0,K + δ [ε v n] p, 3.13) E inter δ div ε v, p) 0,K = δ div ε v, p p) 0,K δ div ε v, p) 0,K, 3.14) 9
δ div ε v, p) 0,K = δ ε v, p) δ [ε v n] p, 3.15) δ ε v, p) = δ ε p, p) = δ ε 1 p 0, 3.16) were, from 3.9), te Poincaré-riedrics inequality, and te uniform ellipticity of ε, we ave δ div ε v, p p) 0,K + δ [ε v n] p p) E ) inter 1/ δ K div ε v 0,K K p p 0,K δ [ε v n] 0, 1 p p 0, δ δ C 3 K div ε v 0,K + K p p 0,K + K div ε v 0,K + K div ε v 0,K + [ε v n] 0, 1 p p 0, [ε v n] 0, p 1 ) [ε v n] 0, δ C 3 C P C 1/ ε ) ε 1 p 0, 3.17) were C 3 C P Cε 1/ ) = CC 4 ). Summarizing 3.11)-3.17) and coosing δ := 1/C4 + C C4), we ave R div ε v) 0, δ [ δc4 + C C4) ] ε 1 p 0 K div ε v 0,K + ) [ε v n] 0, E inter 1 = C4 + C C4 ε 1 p 0 K div ε v 0,K + ) [ε v n] 0,. 3.18) Step. We consider ψ wit ψ 0 in 3.7). We take ψ 0 P 1 )3 as te L projection or te averaging interpolant of ψ 0 H 1 Ω)) 3, suc tat ) 1 K ψ0 ψ 0 0,K + 1 ψ0 ψ 0 0, C 5 ψ 0 1, 3.19) E ψ 0 0 C 5 ψ 0 1. 3.0) Note tat te above ψ 0 P 1 )3 can be constructed from te L projection [39, 38], or from te Clément interpolation in [6], or Bernardi-Girault interpolation [7], or Scott-Zang interpolation [61]. Let δ > 0 be a constant to be determined. We ave R µ 1 curl v) µ = R µ 1 curl v) δ ψ 0 µ δ ψ 0 µ + δ R µ 1 curl v), ψ 0 ) µ. 3.1) 10
Recall, as indicated in te Introduction section, te uniform boundedness of µ wit te constant C µ and te uniform ellipticity of ε wit te constant C ε. rom 3.0) and 3.8), we ave ψ 0 µ C 1/ µ ψ 0 0 C 1/ µ C 5 ψ 0 1 C 1/ µ C 5 C curl ψ 0 C 1/ µ C 5 C C 1/ ε ε 1 curl ψ 0, i.e., putting C 6 := Cµ 1/ C 5 C Cε 1/, we ave ψ 0 µ C 6 ε 1 curl ψ 0, 3.). We ave δ R µ 1 curl v), ψ 0 ) µ = δ curl v, ψ 0 ) δ E Γ v n ψ 0, 3.3) δ curl v, ψ 0 ) = δ curl v, ψ 0 ψ 0 ) + δ curl v, ψ 0 ), 3.4) δ curl v, ψ 0 ) = δ v, curl ψ 0 ) + δ v n ψ 0, 3.5) δ v, curl ψ 0 ) = δ v, curl ψ) = δ ε 1 curl ψ, curl ψ) = δ ε 1 curl ψ 0, 3.6) were, from 3.19), 3.8), and te uniform boundedness and ellipticity of ε wit te same constant C ε see te Introduction section), we ave δ curl v, ψ 0 ψ 0 ) + δ E Γ v n ψ0 ψ 0 ) E Γ = δ curl v, ψ 0 ψ 0 ) 0,K + δ v n ψ0 ψ 0 ) E ) Γ 1 ) 1 δ K ψ 0 ψ 0 0,K K curl v 0,K ) 1 ) 1 δ 1 ψ 0 ψ 0 0, v n 0, E Γ δ C 5 ψ 0 1 E Γ K curl v 0,K + E Γ δ C 5 CCε 1/ ε 1 curl ψ 0 K C K curl v 0,K + v n 0, E Γ v n 0, K curl v 0,K + ) E Γ ) 1 v n 0, δ C 5 CC 1/ ε ) ε 1 curl ψ 0, were C 5 CC 1/ ε ) = C 6 C 1/ µ ). Summarizing 3.1)-3.7) and coosing δ := 1/C 6 + C 6C µ ), we ave 3.7) R µ 1 curl v) µ δ [ δc6 + C6C µ ) ] ε 1 curl ψ 0 K curl v 0,K + ) v n 0, E Γ 1 = C6 + C 6 C ε 1 curl ψ 0 µ K curl v 0,K + E Γ ) v n 0,. 3.8) 11
inally, combining 3.5), 3.18) and 3.8), putting we obtain C 1 := C ε min1/c 4 + C C 4), 1/C 6 + C 6C µ )), C := 1, R div ε v) 0, + R µ 1 curl v) µ C 1 v 0 C 3. Condition number + K div ε v 0,K + [ε v n] 0, + = C 1 v 0 C S v, v). E Γ K curl v 0,K ) v n 0, 3.9) In tis subsection, we sall give te condition number of te resulting linear system. Teorem 3. Assume tat te mes is quasi-uniform as usual. Ten, te condition number of te resulting linear system of problem.14) is O ). Proof. Since R, R are all L projectors, we ave for all v U In fact, from.9) and Proposition 3.3, we ave R div ε v) 0, + R µ 1 curl v) µ C v 0. R div ε v) 0, = ε v, R div ε v)) C v 0 R div ε v) 0 C 1 v 0 R div ε v) 0 C 1 v 0 R div ε v) 0,, were we ave used te inverse estimates q 0 C 1 q 0 for all q P 1 H1 0 Ω). We tus obtain R div ε v) 0, C 1 v 0 v U. rom.10) we ave R µ 1 curl v) µ curl v 0 R µ 1 curl v) 0 + C 1 v 0 R µ 1 curl v) µ + 1 v n 0, E Γ C 1 v 0 R µ 1 curl v) µ, E Γ E Γ v n 0, R µ 1 curl v) 0, R µ 1 curl v) 0, were we ave used te inverse estimates curl v 0 C 1 v 0 for all v U and te trace estimates 1/ v n 0, C 1 K v 0,K and 1/ q 0, C q 0,K for all v U and q P 1)3 and for all K, wit K Γ. Tus, we ave R µ 1 curl v) µ C 1 v 0 v U. Similarly, using te inverse estimates and te trace estimates, we obtain for all v U We ten ave S v, v) C v 0. L v, v) C v 0 v U, wic, togeter wit te L coercivity property 3.1) in Teorem 3.1 and te symmetry property of L, leads to te conclusion of te teorem. 1
4 Error estimates In tis section, we sall estimate te errors between te exact solution and te finite element solution. 4.1 Estimation of consistency errors Here we estimate te consistency errors caused by R, R, S. Lemma 4.1 Let u and u be te exact solution to problem 1.1), 1.), 1.3) and te finite element solution to problem.14), respectively. or all v U, we ave te following consistency formula: L u u, v ) = µ 1 curl u, R µ 1 curl v )) µ µ 1 curl u, curl v ) +div ε u, R div ε v )) div ε u, div ε v ) 0,K + K div ε u, div ε v ) 0,K + K curl u, curl v ) 0,K + v n n µ 1 curl u n) + [ε v n] div ε u. E Γ 4.1) Proof. or te exact solution u, from.9),.11) and Remark.6 we ave R div ε u), R div ε v )) 0, = ε u, R div ε v )) = div ε u, R div ε v )) div ε u, div ε v ) 0,K + div ε u, div ε v ) 0,K. 4.) rom.10) and Remark.6 we ave R µ 1 curl u), R µ 1 curl v )) µ = µ 1 curl u, R µ 1 curl v )) µ = µ 1 curl u, R µ 1 curl v )) µ µ 1 curl u, curl v ) +µ 1 curl u, curl v ), wile we ave from 1.1) for te exact solution u µ 1 curl u, curl v ) + div ε u, div ε v ) 0,K = f, v ) + v n n µ 1 curl u n) + E Γ [ε v n] div ε u. 4.3) 4.4) rom Remark.6 we ave S u, v ) = Kdiv ε u, div ε v ) 0,K + Kcurl u, curl v ) 0,K. 4.5) It follows tat 4.1) olds. Lemma 4. Let u be te exact solution. Ten for all v U we ave K div ε u, div ε v ) 0,K + K curl u, curl v ) 0,K div ε u 0 + curl u 0 ) K C K div ε v 0,K + ) 1 K curl v 0,K. Proof. Tis lemma can be easily proved using te Caucy-Scwarz inequality. Te following proposition is an intermediate step for estimating te consistency error stated in Lemma 4.1, were te involved interpolations div ε u P 1 H1 0 Ω) and µ 1 curl u P 1)3 to div ε u and µ 1 curl u 13
can be obtained from te Clément interpolation [6], Bernardi-Girault interpolation [7], or Scott-Zang interpolation [61]. Te latter can also be obtained from te L projection [39, 38]. Proposition 4.1 Let div ε u P 1 H1 0 Ω) and µ 1 curl u P 1)3 denote te interpolants of div ε u and µ 1 curl u, respectively, were u is te exact solution. or all v U we ave µ 1 curl u, R µ 1 curl v )) µ µ 1 curl u, curl v ) = µ 1 curl u µ 1 curl u, R µ 1 curl v )) µ + µ 1 curl u µ 1 curl u, curl v ) v n n µ 1 curl u n); E Γ div ε u, R div ε v )) div ε u, div ε v ) 0,K = div ε u div ε u, R div ε v )) + div ε u, R div ε v )) div ε u, R div ε v )) 0, + div ε u div ε u, div ε v ) 0,K [εv n] div ε u. Proof. Equation 4.6) olds, since from.10) we can obtain µ 1 curl u, R µ 1 curl v )) µ = µ 1 curl u, curl v ) v n n µ 1 curl u n). 4.8) Equation 4.7) olds, since div ε u, R div ε v )) = div ε u, R div ε v )) div ε u, R div ε v )) 0, + div ε u, R div ε v )) 0,, 4.9) were from.9) we ave E Γ div ε u, R div ε v )) 0, = ε v, div ε u) = div ε u, div ε v ) 0,K 4.6) 4.7) [ε v n] div ε u. 4.10) In order to estimate te consistency error in 4.1), from 4.6) and 4.7) it is obvious tat divεu and µ 1 curlu must ave suitable regularity. Tis issue is dealt wit as follows. Introduce te div Hilbert spaces Hdiv; µ; Ω) = {v L Ω)) 3 : div µ v L Ω)}, H 0 div 0 ; µ; Ω) = {v Hdiv; µ; Ω) : div µ v = 0, µ v) n Γ = 0}. Let u be te exact solution of 1.1), 1.) and 1.3), and let z := µ 1 curl u Hcurl; Ω) H 0 div 0 ; µ; Ω), p := div ε u H 1 0 Ω). Ten, assuming a suitable smoot f, by obvious manipulations, from 1.1)-1.3) we can ave te following two elliptic and Maxwell s interface problems: div ε p = div f in Ω l 1 l L, p Γ = 0, [p] = 0, [ε p n] = [f n] on Γ ij, Hypotesis H1) curl ε 1 curl z = curl ε 1 f, div µ z = 0 in Ω l 1 l L, µ z) n Γ = 0, ε 1 curl z n Γ = ε 1 f n Γ, [z n] = 0, [µ z n] = 0, [ε 1 curl z n] = [ε 1 f n] on Γ ij. or a suitable smoot f, we assume tat p L H1+r Ω l ) wit r > 1/, satisfying L p 1+r,Ωl C f). were C f) is a constant depending on f. In te following remark, we provide some cases were C f) can be explicitly given. 14
Remark 4.1 or a globally smoot ε say ε = 1) and f Hdiv; Ω), Hypotesis H1) olds see [40]), wit C f) C divf r 1 for some r > 1/. or a piecewise smoot ε as assumed, te elliptic interface problem wic p solves belongs to te following general elliptic interface problem: given g and χ, to find θ suc tat div ε θ = g in Ω l 1 l L, θ Γ = 0, [θ] = 0, [ε θ n] = χ on Γ ij. or L = i.e., Ω = Ω 1 Ω ), if Ω 1 Ω and Ω = Ω \ Ω 1, wit te material interface Γ 1 = Ω 1 Ω, and if ε is a piecewise constant i.e., ε Ωl = c l, were c l is constant), ten from Remark.4 in [17] tere exists a r > 1/ suc tat L θ 1+r,Ωl C g r 1,Ω + χ r 1/,Γ1 ), were g H r 1 Ω) and χ H r 1/ Γ 1 ). In our case, we may ten require tat divf H r 1 Ω) and f Ωl H r Ω l )) 3 for some r > 1/, and C f) C divf r 1,Ω + L f r,ω l ), were we used te trace teorem: H r Ω l ) is continuously embedded into H r 1/ Ω l ). or piecewise smoot material interfaces, more general piecewise regularity estimates for te above elliptic interface problems may be referred to [58, 46, 1]. or Hypotesis H1), from [17, 58, 46, 1] and from Teorem 4.1 in [31], a reasonable regularity on f seems to be tat f L Hdiv; Ω l) L Hr Ω l )) 3 for some r > 1/. Wit tis f and te assumed piecewise smoot material coefficient ε i.e., ε Ωl W 1, Ω l )) 3 3 for 1 l L), since [f n] Γij H r 1/ Γ ij ), under te assumptions of Teorem 4.1 in [31] or under te assumptions of Remark.4 in [17], we could expect tat Hypotesis H1) olds, wit C f) C L divf 0,Ω l + f r,ωl ). Tis issue is owever rater complicated in tree-dimensions and is beyond te scope of tis paper, so we do not intend to deal wit it ere. Hypotesis H) can be written into te following regular-singular decomposition or a suitable smoot f, we assume tat z L Hr Ω l )) 3 wit r > 1/ and tat z z = z H + φ in Ω l, 1 l L, were z H Hcurl; Ω) L H1+r Ω l )) 3 and φ H 1 Ω) L H1+r Ω l ) satisfy L z H 1+r,Ωl + φ 1+r,Ωl ) C f), were C f) is a constant depending on f. Remark 4. Suc regular-singular decompositions in Hypotesis H) for Maxwell interface problems can be found in [57, 31, 30, 9]. In te case were µ = ε = 1, or µ and ε are globally in W 1, Ω)) 3 3, from [57] we conclude tat Hypotesis H) olds, wit C f) C curlε 1 f 0 + f r ). In general, from Teorem 7.3 in [31] we could assume a reasonable regularity on f for Hypotesis H) to be true, i.e., f L Hcurl; ε 1 ; Ω l ) L Hr Ω l )) 3 for some r > 1/, were Hcurl; ε 1 ; Ω l ) = {v L Ω l )) 3 : curl ε 1 v L Ω l )) 3 }. or tis f and te assumed piecewise smoot material coefficients µ and ε i.e., µ Ωl, ε Ωl W 1, Ω l )) 3 3, 1 l L), since ε 1 f n belongs to H r 1/ Γ)) 3 and [ε 1 f n] belongs to H r 1/ Γ ij )) 3, under te assumptions of Teorem 7.3 in [31], we could expect tat Hypotesis H) olds, wit C f) C L curlε 1 f 0,Ωl + f r,ωl ). Proposition 4. See [6]) or te mass-lumping L inner product.8), we ave p, q) 0 p, q) 0, C p 1 q 0 p, q P 1 H 1 0 Ω). Lemma 4.3 Let u be te exact solution. Assuming Hypotesis H1), we ave div ε u, R div ε v )) div ε u, div ε v ) 0,K + [ε v n] div ε u C R div ε v ) 0, + C r K div ε v 0,K + 15 [ε v n] ) 1.
Proof. According to p = div ε u in Hypotesis H1), we define p P 1 H1 0 Ω) te interpolant to p, e.g., see [39, 6, 7, 61, 4, 14], suc tat K p p 0,K + K 1 p L p 0, C r p 1+r,Ωl, p 1 C p 1, p p 0 C p 1. Ten, from Lemma 4.1 and 4.7) in Proposition 4.1 and Proposition 4., we ave div ε u, R div ε v )) div ε u, div ε v ) 0,K + [ε v n] div ε u = p, R div ε v )) p, R div ε v )) 0, + p p, R div ε v )) + p p, div ε v ) 0,K + [εv n] p p), 4.11) were p, R div ε v )) p, R div ε v )) 0, C p 1 R div ε v ) 0 C R div ε v ) 0,, p p, R div ε v )) p p 0 R div ε v ) 0 C R div ε v ) 0 C R div ε v ) 0,, p p, div ε v ) 0,K C r K p p 0,K K div ε v 0,K K div ε v 0,K Lemma 4.4 [εv n] p p) C r 1 p p 0, [εv n]. Let u be te exact solution. Assuming Hypotesis H), we ave [εv n] µ 1 curl u, R µ 1 curl v )) µ µ 1 curl u, curl v ) + v n n µ 1 curl u n) E Γ C r R µ 1 curl v ) µ + C r K curl v 0,K + E Γ v n ) 1. 4.1) Proof. According to z = µ 1 curl u = z H + φ in Hypotesis H), we define z P 1 )3 as follows: z = z H + φ, were z H P 1 )3 is te interpolant to z H and φ P 1 H1 Ω) is te interpolant to φ, see [6, 7, 39, 61, 4, 14] for general finite element interpolation teory, satisfying K z H z H 0,K + K 16 1 z H z H 0, + φ φ 1 C r,
and we ave We also ave z z 0 C r. φ φ), curl v ) = v n n φ φ) n) v U. E Γ Terefore, we ave from 4.6) in Proposition 4.1 and Lemma 4.1 tat µ 1 curl u, R µ 1 curl v )) µ µ 1 curl u, curl v ) + v n n µ 1 curl u n) = z z, R µ 1 curl v )) µ + z H z H, curl v ) + φ φ), curl v ) + v n n z H z H ) n) + v n n φ φ) n) E Γ E Γ = z z, R µ 1 curl v )) µ + z H z H, curl v ) + v n n z H z H ) n), were z z, R µ 1 curl v )) µ z z µ R µ 1 curl v ) µ C r R µ 1 curl v ) µ, E Γ E Γ and E Γ z H z H, curl v ) C r v n n z H z H ) n) K z H z H 0,K K curl v 0,K E Γ C r 1 z H z H 0, E Γ K curl v 0,K v n. E Γ v n It follows tat 4.1) olds. Simply summarizing tese lemmas from Lemma 4.1 to Lemma 4.4 we obtain te estimate for te consistency errors caused by L projections and te stabilization in te following teorem. Teorem 4.1 Let u and u be te exact solution of problem 1.1), 1.), 1.3) and te finite element solution of problem.14), respectively. Ten, under Hypoteses H1) and H), L u u, v ) C r v L v U, were v L := L v, v ). 4.13) 4. Constructing an interpolant of te solution In tis subsection we sall construct a finite element interpolant ũ U of te solution u, wic is to be used in next subsection to estimate te error of our continuous finite element metod. Teorem 4. Let u be te exact solution and u L Hr Ω l )) 3 wit r > 1. Let U be defined as in.6). Ten, tere is a function ũ U suc tat R div ε u ũ)) = 0, R µ 1 curl u ũ)) = 0 4.14) 17
and u ũ 0 C r L u r,ωl. 4.15) Proof. irstly tere is a function u 0 U suc tat See [61, 6, 7, 6, 14, 7]) u u 0 0 + K We ten define ũ U by te following conditions: and According to.3)-.6) we can write ũ in K as ũ = u 0 + L u u 0 0, C r u r,ωl. 4.16) ũa) = u 0 a) for all vertices a, 4.17) ũ u) p = 0 p P 1 )) 3, K, wit Γ, K 4.18) K Γ m K ũ u) q = 0 q P 0 ε; K). 4.19) m K m K c,l p,l b + c K,l q K,l b K =: û + c K,l q K,l b K, 4.0) were P 1 )) 3 = span{p,l, were p,l K P 1 K)) 3, 1 l m }, in wic te integer m denotes te dimension of P 1 )) 3 namely, m = 9), and P 0 ε; K) = span{q K,l, 1 l m K }, and c,l, c K,l R are all coefficients to be determined according to 4.18) and 4.19), i.e. and m c,l m K c K,l p,l p,i b = K q K,l q K,i b K = u u 0 ) p,i 1 i m, K 4.1) K u û) q K,i 1 i m K, K. 4.) Since te bubble-related terms in 4.0) are zero at any vertex and te element bubble terms are zero along any face, we can obtain tat ũ U is uniquely determined by 4.17)-4.19). Using te standard scaling argument See e.g. [4]), we can ave and u û 0,K C u u 0 0,K + C K It follows from 4.3), 4.4) and 4.16) tat 4.15) olds. Now we sall verify 4.14). By virtue of 4.19) and.9), we ave 1 u u 0 0, 4.3) u ũ 0,K C u û 0,K. 4.4) R div ε u ũ)) 0, = ε u ũ), R div ε u ũ))) = u ũ, ε R div ε u ũ))) = 0 4.5) 18
since ε R div ε u ũ)) K εp 0 K)) 3 P 0 ε; K), wile from 4.18) and.10), we ave R µ 1 curl u ũ)) µ = u ũ, curl R µ 1 curl u ũ))) 0,K + u ũ) n R µ 1 curl u ũ)) = 0 K E Γ 4.6) since curl R µ 1 curl u ũ)) K P 0 K)) 3 P 0 ε; K) and n R µ 1 curl u ũ)) P 1 )) 3. So 4.14) olds. Remark 4.3 Te assumption tat te exact solution u L Hr Ω l )) 3 for some r > 1/ is commonly used in te literature, see, e.g., [], wile in [44] even more regularity is assumed, i.e., r = 1, and in [3] a piecewise regularity W 1,p Ω l ) for some p > is assumed. Meanwile, we ave known tat r = 1 is sown for te case were two materials occupy a convex polyedron Ω in [46]. We sould point out tat, from Sobolev embedding teorem [1, 39], te main reason for r > 1/ is tat it makes te element-face integrals of u well-defined, and te oter reason is tat r > 1/ ensures tat te traces of f in Hypoteses H1) and H) are well-defined. Remark 4.4 Te tecnique for constructing te finite element interpolant from equations 4.17)-4.19) is quite classical, e.g., see Lemma A.3 in [39], and somewat similar tecnique may be also found in Section, Capter II of [39]. 4.3 Error estimates in an energy norm We introduce an energy norm: v,curl,div : = v 0 + v L = v 0 + L v, v) = v 0 + R div ε v) 0, + R µ 1 curl v) µ + S v, v). 4.7) Recall Ω = L Ω l, were {Ω l, 1 l L} is te partition of Ω associated wit discontinuities of ε. Let E l denote te collection of element faces in Cl, te subset of C corresponding to Ω l. On Ω l, let V be te Crouzeix-Raviart nonconforming linear element [3], i.e., { } V = z L Ω l ); z K P 1 K), [z] = 0, E l. 4.8) Let J : H r Ω l ) V, r > 1/ be te interpolation operator, for z we define J z V by J z = z E, l 4.9) and let J : H r Ω l )) 3 V ) 3 be te vector interpolation operator. Let Π 0 : L Ω l ) P 0 denote te piecewise-constant L projection, i.e., for z we ave K Π 0 z K = z K C l K, 4.30) and Π 0 : L Ω l )) 3 P 0)3 te vector L projection. Proposition 4.3 See [3]) Given z H r Ω l ) or z H r Ω l )) 3. We ave te following relations: J z z 0,K + 1/ J z z 0, C r K z r,k K C l, div J z = Π 0 div z, divj z z) 0,K div z 0,K K C l, curl J z = Π 0 curl z, curlj z z) 0,K curl z 0,K K C l. 19
Lemma 4.5 Recall tat ε = ε ij ) is piecewise smoot, i.e., ε ij Ωl W 1, Ω l ), 1 l L. or te exact solution u and its interpolation ũ U as in Teorem 4., we ave S u ũ, u ũ) C r div ε u 0 + curl u 0 + L u r,ωl ). 4.31) Proof. Let ε u := J ε u V ) 3 be te nonconforming linear element interpolation of ε u Ωl. Under te regularity assumption on ε, we ave from Proposition 4.3 tat for any K C l ε u ε u 0,K + 1/ ε u ε u 0, C r K u r,k, div ε u ε u) 0,K C div ε u 0,K. In addition, let ε 0 be some finite element interpolation of ε, satisfying [4] ε ε 0 0,,K + K ε ε 0 1,,K C K ε 1,,K, were 0,,K, 1,,K denote te semi-norms of Sobolev space W 1, K), see [1]. We first consider te error term. K div ε u ũ) 0,K We ave were K div ε u ũ) 0,K K div ε u ε u) 0,K + K div ε u ε 0 ũ) 0,K + K div ε 0 ε) ũ 0,K, K div ε u ε u) 0,K C K div ε u 0,K, K div ε u ε 0 ũ) 0,K C ε u ε 0 ũ 0,K C ε u ε u 0,K + ε u ũ) 0,K + ε ε 0 ) ũ u) 0,K + ε ε 0 ) u 0,K ), K div ε 0 ε) ũ 0,K C K ε 0 ε 1,,K ũ 0,K + C K ε 0 ε 0,,K ũ 1,K, K ε 0 ε 1,,K ũ 0,K K ε 0 ε 1,,K ũ u 0,K + K ε 0 ε 1,,K u 0,K, K ε 0 ε 0,,K ũ 1,K C ε 0 ε 0,,K ũ 0,K C ε 0 ε 0,,K ũ u 0,K + C ε 0 ε 0,,K u 0,K. By collecting all above estimates we obtain C K div ε u ũ) 0,K L div ε u 0,Ωl + ε u ε u 0,Ωl + ũ u 0,Ωl ) +C L ε ε 0 0,,Ωl + ε ε 0 1,,Ωl ) u 0,Ωl ) C r div ε u 0 + L u r,ωl. Regarding te error term 1/ [εu ũ) n], 0
wit two sides +, of, we ave [εu ũ) n] C εu ũ) 0, + C + εu ũ) 0,, were for eiter side of wit te corresponding element K we ave and 1/ εu ũ) 0, 1/ εu ε u 0, + 1/ ε u ε ũ 0,, 1/ ε u ε ũ 0, 1/ ε u ε 0 ũ 0, + 1/ ε 0 ũ ε ũ 0,, 1/ ε u ε 0 ũ 0, C ε u ε 0 ũ 0,K, 1/ ε 0 ũ ε ũ 0, C ε 0 ũ ε ũ 0,K + K ε 0 ũ ε ũ 1,K ) ε 0 ũ ε ũ 1,K C ε 0 ε 1,,K ũ 0,K + ε 0 ε 0,,K ũ 1,K ). Ten, all te subsequent estimations are te same as tose for te error term 1/, K div ε u 0,K) ũ) we tus obtain 1/ ) L [εu ũ) n] C r div ε u 0 + u r,ωl. u ũ) n, following simi- To estimate K curl u 1/ 0,K) ũ) and E Γ lar arguments as above, wit û := J u V ) 3 we ave K curl u ũ) 0,K + E Γ u ũ) n 1/ C r curl u 0 + L u r,ωl ). Remark 4.5 If u is in H 1 Ω)) 3, ten from ũ 1 C u 1 we easily ave S u ũ, u ũ) C u 1, wic is consistent wit 4.31). In tat case, similar error estimates ave been known in te stabilization finite element metods in te literature e.g. [37]). However, for u not in H 1 Ω)) 3 but in H r Ω)) 3 wit r < 1, te estimate in Lemma 4.5 is new, to te best of te autors knowledge. Teorem 4.3 Let u and u be te solution of problem 1.1), 1.), 1.3) and te finite element solution of.14), respectively. Assume tat Hypoteses H1) and H) old and tat u L Hr Ω l )) 3 wit r > 1/. Ten u u,curl,div C r. 4.3) Proof. rom te symmetry and coercivity properties of L, we ave te Caucy-Scwarz inequality: L u, v) L u, u)) 1 L v, v)) 1. rom Teorem 4.1 we ave u ũ L = L u ũ, u ũ) = L u ũ, u ũ) + L u u, u ũ) u ũ L u ũ L + C r u ũ L C u ũ L + r ) u ũ L, 1
tat is, were, from Teorem 4. and Lemma 4.5, u ũ L C u ũ L + r ), 4.33) u ũ L = L u ũ, u ũ) = R div ε u ũ)) 0, + R µ 1 curl u ũ)) µ + S u ũ, u ũ) = S u ũ, u ũ) C r div ε u 0 + curl u 0 + L u r,ωl ). 4.34) Terefore, by te triangle inequality, from Teorem 3.1 and 4.34) and Teorem 4., we ave u u 0 u ũ 0 + u ũ 0 u ũ 0 + C u ũ L u ũ 0 + C u ũ L + r ) C u ũ 0 + C r C r, 4.35) and by te triangle inequality again, from 4.33) and 4.34), we ave u u L u ũ L + ũ u L C r. 4.36) Summarizing 4.35), 4.36) and 4.7), we ave 4.3). Remark 4.6 rom subsection 4.1 te constant in 4.3) would take te form div ε u 0 + curl u 0 + L u r,ωl + L divf 0,Ω l + curl ε 1 f 0,Ωl + f r,ωl ) up to a multiplicative constant wic is independent of u, f and. On te oter and, as remarked in Remark.9, for te first-order system like 1.9), since no consistency errors exist, from te above argument te constant in 4.3) takes te form div ε u 0 + curl u 0 + L u r,ωl, up to a multiplicative constant wic is independent of u, f and. Remark 4.7 or piecewise non-polynomial ε, we may consider to replace ε by a suitable piecewise polynomial approximation, say ε see below). Wit tis ε, all te previous analysis and results are almost uncanged, except te additional consistency error terms caused by tat replacement as follows: and div ε ε ) u, R div ε v )), 4.37) [ε ε ) u n] [ε v n] 4.38) K div ε ε ) u, div ε v ) 0,K. 4.39) Recall ε ij L W 1, Ω l ), 1 i, j 3. We coose ε l in Ω l 1 l L) as a continuous finite element approximation of ε Ωl wit ε l ij, K P1 K), were P1 K) denotes te linear element enriced by tree face bubbles and one element bubble. Recall tat K is a tetraedron. rom [7, 6, 4, 39] we ave ε ε 0,,K + K ε ε 1,,K C K ε 1,,K for all K C, were 1, denotes te semi-norms of W 1,. Note tat ε satisfies te same uniform ellipticity property as ε for a sufficiently small. Also, note tat te bubbles in P1 K) are introduced in order tat all ε l ij,, 1 i, j 3, satisfy te following interpolation properties: ε l ij ε l ij,) q = 0 q P 1 ), K, K C 4.40) and K ε l ij ε l ij,) = 0 K. 4.41)
We sall only explain ow to estimate 4.37) below. Oter error terms can be estimated similarly. On eac face K for any K C, let ū := 1 u. We ave u ū 0, C r 1 K u r,k. Letting K û := u K, we ave u û 0,K C r K u r,k. Ten, from 4.40), 4.41) and te interpolation properties of ε, ū and û, we ave div ε ε ) u, R div ε v )) = ε ε ) u, R div ε v )) + ε ε ) u n R div ε v ) = ε ε ) u û), R div ε v )) K + ε ε ) u ū) n R div ε v ) C r R div ε v ) 0,, K 4.4) were C depends on u r, L ε 1,,Ω l, but it is independent of. 5 Conclusion In tis paper we ave proposed a continuous finite element metod to solve te curlcurl-grad div interface problem in a tree-dimensional polyedron domain. Different anisotropic nonomogeneous materials are allowed to occupy multiple subdomains of te domain, wit te two caracteristic coefficients being discontinuous from subdomain to subdomain. Due to spatial inomogeneities, suc interface problems are frequently encountered in many pysical science and engineering applications suc as electromagnetism and fluid-structure interaction. Te key feature of te metod proposed in tis paper is te application of two suitable L projections to curl and div operators. Wit respect to te regularity associated wit te problem in material subdomains, te error bound O r ) in an energy norm between te exact solution and te continuous finite element solution is proved. rom section 4 of tis paper, te regularity requirement of te error bound O r ) can be generally summarized as follows: u H 0 curl; Ω) Hdiv; ε; Ω) L Hr Ω l )) 3, µ 1 curlu Hcurl; Ω) H 0 div; µ; Ω) L Hr Ω l )) 3, ε, µ L Ω)) 3 3 L W 1, Ω l )) 3 3, and f L Ω)) 3 L Hdiv; Ω l) Hcurl; ε 1 ; Ω l ) H r Ω l )) 3, were r > 1/ and te regularity on f could ensure te regularity of µ 1 curlu, as stated in Hypotesis H). On te oter and, if problem 1.1) comes from a first-order system like 1.9), ten u H 0 curl; Ω) Hdiv; ε; Ω) L Hr Ω l )) 3 for r > 1/ and ε L Ω)) 3 3 L W 1, Ω l )) 3 3 are sufficient. Te continuous finite element metod developed in tis paper is a general-purpose metod and as broad applications, since it can work no matter if te general tensor coefficients are discontinuous or not, and no matter if te exact solution as a low regularity or not, and no matter if te continuous problem comes from Maxwell s second-order or first-order curl/ div interface problems or from oter similar interface problems involving curl/div operators. Acknowledgments Te autors would like to express teir gratitude to te anonymous referees for teir valuable comments and suggestions wic ave elped to improve te overall presentation of te paper. References [1] A. Adams: Sobolev Spaces, Academic Press, New York, 1975. [] A. Alonso and A. Valli: Some remarks on te caracterization of te space of tangential traces of Hrot; Ω) and te construction of an extension operator, Manuscr. Mat., 89 1996), 159-178. [3] C. Amrouce, C. Bernardi, M. Dauge, and V. Girault: Vector potentials in tree-dimensional nonsmoot domains, Mat. Metods Appl. Sci., 1 1998), 83-864. 3