A THIRD ORDER LINEARIZED BDF SCHEME FOR MAXWELL S EQUATIONS WITH NONLINEAR CONDUCTIVITY USING FINITE ELEMENT METHOD

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 4, Number 4-5, Pages 5 5 c 7 Institute for Scientific Computing and Information A THIRD ORDER LINEARIZED BDF SCHEME FOR MAXWELL S EQUATIONS WITH NONLINEAR CONDUCTIVITY USING FINITE ELEMENT METHOD CHANGHUI YAO, YANPING LIN, CHENG WANG, AND YANLI KOU Abstract. In tis paper, we study a tird order accurate linearized backward differential formula (BDF) type sceme for te nonlinear Maxwell s equations, using te Nédelec finite element approximation in space. A purely explicit treatment of te nonlinear term greatly simplifies te computational effort, since we only need to solve a constant-coefficient linear system at eac time step. An optimal L error estimate is presented, via a linearized stability analysis for te numerical error function, under a condition for te time step, C for a fixed constant C. Numerical results are provided to confirm our teoretical analysis and demonstrate te ig order accuracy and stability (convergence) of te linearized BDF finite element metod. Key words. Maxwell s equations wit nonlinear conductivity, convergence analysis and optimal error estimate, linearized stability analysis, te tird order BDF sceme.. Introduction () () Tis paper is concerned wit te nonlinear Maxwell s equations wit initial and boundary conditions () (4) ǫe t +σ(x, E )E H =, in Ω (,+ ), µh t + E =, in Ω (,+ ), E(x,) = E (x), H(x,) = H (x), in Ω, n E = and n H =, on Γ (,+ ), were Ω is a bounded, convex, simply-connected domain in R wit a regular boundary Γ = Ω, E(x,t), H(x,t) represent te electric and magnetic fields, n is te outward normal vector on Γ, and te positive constants ǫ and µ stand for te permittivity and te magnetic permeability, respectively. In addition, σ = σ(x, s) is a real valued function representing te electric conductivity. Te system() (4) ave been investigated in [6,, ]. Te autors proved te existence of te weak solution for a nonlinear function J(E) = σ( E )E, wit σ(s) monotonically increasing. In[4], te autors presented te existence and uniqueness of te sceme by discretizing te time domain and taking te limit for infinitely small time-step. In section 5 of [4], te autors also proved tat te solutions converges to te quasi-state state (no time derivative for equation involving E) wen te permittivity ǫ. Also wen ǫ goes zero, numerical example indicated tat numerical sceme converges to te quasi static state as part of verification. In [5], te autors proved existence and uniqueness of te discrete fields using te monotone operator teory (see [7]). In [6], te autors studied a time dependent eddy current equation, establised te existence and uniqueness of a weak solution in suitable function space, designed a nonlinear time discrete approximation sceme Received by te editors January, 7 and, in revised form, Marc, 7. Matematics Subject Classification. 5R5, 49J4, 6G4. 5

2 5 C. YAO, Y. LIN, C. WANG, AND Y. KOU based on te Rote s metod and proved te convergence of approximation to a weak solution. Numerical analysis for te nonlinear system ave also been extensively carried out, see [,, 4, 5, 9,,,,, 4, 8,,, 5, 6]. In addition, nonlinear scemes ave been proposed and analyzed in many literatures. In [4], te autors presented a numerical sceme to solve coupled Maxwell s equations wit a nonlinear conductivity, wit te backward Euler discretization in time and mixed conforming finite elements in space. And also, a mixed finite element metod for te Maxwell s equations wit a nonlinear boundary condition was studied in [5]. In [], te autors proposed a fully-discrete finite element metod to solve te time-domain metamaterial Maxwell s equations, wic can be reduced to a vector wave integrodifferential equation involving just one unknown. Some related works can also be found in [9,,, 8]. In [5], te autors proposed a numerical sceme based on backward Euler discretization in time and curl-conforming finite element in space to solve Maxwell s equations wit nonlinear conductivity in te form of a power law. As a result, its convergence was proved, based on te boundedness of te second derivative in te dual space by te Minty-Browder tecnique. In [], te autors developed a fully-discrete (T,ψ) ψ e finite element decoupled sceme to solve time-dependent eddy current problem wit multiple-connected conductors. Subsequently, an improved convergence rate analysis was provided in [4]. A few more earlier works are also available in [, 6]. Clearly, linearized scemes are muc more efficient tan nonlinear scemes for solving nonlinear equations, since only one linear system solver is needed in te former one, wile te latter one always requires a nonlinear iteration solver at eac time step. For example, a new approac was developed in [6], based on a temporal-spatial error splitting tecnique by introducing a corresponding timediscrete system. Similarly, a linearized backward differential formula (BDF) type sceme was applied to te time-dependent nonlinear termistor equation in [7]. Te linearized backward Euler sceme for te nonlinear Joule eating equation was studied in [8]. In [9], te autors presented an optimal L error estimate of a linearized Crank-Nicolson Galerkin FEM for a generalized nonlinear Scrödinger equation, witout any time step size restriction. In turn, an important question arises: Is it possible to design iger order ( ) linearized temporal discretization for te nonlinear Maxwell s equations, wit a convergence analysis available? In tis paper, we give an affirmative answer to tis question. We propose a tird order accurate, linearized BDF type FEM metod for te Maxwell s equations and provide a teoretical error analysis for te proposed sceme. It is well-known tat te rd order BDF temporal sceme is not A-stable; in fact, its stability domain does not contain any part on te purely imaginary axis, were all te eigenvalues of te linear Maxwell operator are located. Tis fact makes a teoretical analysis for te rd order BDF metod applied to te Maxwell equation igly callenging. To overcome tis subtle difficulty, we take an inner product wit te numerical error equation by e n+ + (λ + )(e n+ e n ) (wit λ > and e k denoted as te numerical error function at time step t k ), and employ a telescope formula establised in []. Moreover, due to te yperbolic nature of te Maxwell s equation, a requirement for te time step size, C (wit C a fixed constant), as to be imposed to pass troug te numerical error estimate. Anoter key tecnical contribution in te convergence analysis is to obtain an L bound of te numerical solution, via a linearized stability analysis for te numerical error function; by contrast, suc an L bound was not needed in te previous

3 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION 5 works [5, 4,, 6], due to te implicit treatment of te nonlinear term. In more details, an a-priori L assumption is made for te numerical error function at te previous time steps, and suc an L bound for te numerical solution leads to a full order O( + s ) error estimate at te next time step. In turn, an application of inverse inequality yields te L bound of te numerical error function at te next time step, so tat an induction could be applied to complete te nonlinear error estimate, under a trivial requirement o( d 6 ). Similar metodology of linearized stability analysis for nonlinear problems could also be found in a few related works [5, 6, 8, 9], etc. Of course, te time step requirement in te nonlinear error estimate, o( d 6), is muc less severe tan te one imposed for te linear Maxwell part, C, associated wit te stability analysis for te rd order BDF temporal stencil. As a result, te more severe time step constraint, C, is needed to pass troug te convergence analysis for te wole numerical sceme. Meanwile, suc a severe time step constraint is only a teoretical issue; extensive numerical results ave sown tat, te stability and convergence are well preserved wit a more relaxed constraint, = O(). Tis analysiscouldeasilygotrougfor apolynomialelement wit degrees, since te O( s ) spatial convergence order in L norm enables one to obtain an L bound of te numerical error function. For a linear element wit s =, suc an argument is not directly available. Instead, we perform a super-convergence of te lowest element over a uniform mes, so tat an O( ) spatial convergence in L is valid. In turn, an application of inverse inequality yields te desired L bound for te numerical solution. Te rest of te paper is organized as follows. In Section., we present te linearized BDF finite element metod and state te main teoretical result. In Section., we derive an L error estimate of te fully discrete system and get te corresponding estimate of order O( + s ), under a requirement C. In particular, te super-convergence analysis for s = is also provided in tis section. In Section. 4, numerical results are presented to demonstrate te teoretical analysis. Finally, te concluding remarks are provided in Section. 5.. Finite element metod and te main teoretical result Te inner product in [L (Ω)] will be denoted by (u,v) = u vdx, and te Ω corresponding norm is given by u = (u,u). In tis article, we use te standard Sobolev spaces and introduce some common notation: H(curl,Ω) = {u [L (Ω)] ; u [L (Ω)] }, H (curl,ω) = {u H(curl,Ω);n u = on Γ}, H(div,Ω) = {u [L (Ω)] ;div u L (Ω);div u = }. Here, we need to employ te assumptions of σ(x,s) in [, ]. And also, it is assumed tat tere exist positive constants κ,σ > wit < κ σ(x,s) σ and σ(x,s) C, (Ω,[,+ )). Te problem () (4) as a unique solutions (E,H) suc tat [6] E L (, ;H (curl,ω)) H(div,Ω) L p+ (, ;[L p+ (Ω)] ), H L (, ;H (div,ω)) H (, ;[H (Ω)] ). To introduce te mixed FEM, we partition Ω by a family of regular cubic meses T wit maximum mes size. For te spatial approximation, we employ D cubic

4 54 C. YAO, Y. LIN, C. WANG, AND Y. KOU Raviart-Tomas-Nédelec elements [, ] V = {φ H(curl,Ω),φ K Q k,k,k Q k,k,k Q k,k,k, K T }, W = {ψ [L (Ω)],ψ K Q k,k,k Q k,k,k Q k,k,k, K T }, or D tatraedron Raviart-Tomas-Nédelec elements V = {φ H(curl,Ω),φ K P k,k,k P k,k,k P k,k,k, K T }, W = {ψ [L (Ω)],ψ K P k,k,k P k,k,k P k,k,k, K T }. In turn, we denote te interpolation operator Π and projection operator P on V and W, respectively. Te following important properties could be found in [,,, 8]. Lemma.. For te space pairs V and W, tere olds (5) curl V W. If v is a function suc tat bot te interpolants Π v and P (curlv) exist, ten we get curl (Π v) = P (curlv). For w [L (Ω)], te following property olds: (w P w,q) =, q W. We need te optimal interpolation error estimate[,, ] for general anisotropically refined meses. For regular meses, a similar result is obtained as follows. Lemma.. For s k,v H s+ (Ω),w H s (Ω), tere exists a constant C > independent of suc tat v Π v H(curl,Ω) C s v s+, w P w C s w s, were s denotes te seminorm in te space H s (Ω). For t (,T], te weak formulation of () () wit te boundary condition (4) is defined by (6) (7) (ǫe t,ξ u )+(σ(x, E )E,ξ u ) (H, ξ u ) =, ξ u V, (µh t,ξ φ )+( E,ξ φ ) =, ξ φ W. Moreover, let {t n } N n= be a partition in time direction wit t n = n,t = N and E n = E(x,t n ), H n = H(x,t n ). For any sequence of functions {f n } N n=, te tird order BDF temporal discrete operator is defined as (8) D f n = ( 6 fn f n + fn fn ), n N. Now, we turn our attention to te computation of te starting approximations. Wen n =, we define (9) () E = Π E, H = P H. In order to compute (E,H ), let (Ẽ, H ) be te solution of te following equations () () Ẽ (ǫ E,ξ u ) ( H, ξ u)+(σ(x, E )E,ξ u ) =, ξ u V, H (µ H,ξ φ )+( Ẽ,ξ φ) =, ξ φ W.

5 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION 55 Afterward, (E,H ) (corresponding to n = ) is updated as () (ǫ E E,ξ u ) ( H +H, ξ u )+ (σ(x, Ẽ )Ẽ +σ(x, E )E,ξ u ) =, ξ u V, (4) (µ H H Next, for n =, (E,H ) is determined by,ξ φ )+( ( E +E ),ξ φ ) =, ξ φ W. (5) (ǫ E E,ξ u ) ( H +H, ξ u )+ (σ(x, E )E σ(x, E )E,ξ u) =, ξ u V, (6) (µ H H,ξ φ )+( ( E +E ),ξ φ ) =, ξ φ W. Wit te above numerical solutions at te first two time steps, we introduce te linearized tird order BDF finite element metod for te nonlinear Maxwell s equations () (4): find E n V,H n W,n, suc tat (7) (8) (ǫd E n,ξ u ) (H n, ξ u )+(σ(x, E n )E n +σ(x, E n )E n,ξ u ) =, ξ u V, n, (µd H n,ξ φ )+( E n,ξ φ ) =, ξ φ W. σ(x, E n )E n Lemma.. At eac time step, te system (9) (8) is uniquely solvable. Proof. Since te linear system generated by (7) and (8) is square, te existence of te solution is implied by its uniqueness. Te omogeneous part of sceme (7) (8) is given by (9) () ǫ 6 (En,ξ u) (H n, ξ u) =, ξ u V, µ 6 (Hn,ξ φ)+( E n,ξ φ) =, ξ φ W. By coosing ξ u = E n,ξ φ = H n in (9) and (), we ave () ǫ 6 En + µ 6 Hn =, wic in turn yields E n =,Hn =. Te existence and te uniqueness of(9) (6) can be similarly proved. Tis completes te proof of Lemma.. In te rest part of tis paper, te following regularity is assumed for te exact solution of te initial boundary value problem () (4): () () E Hs (Ω) + E t L (,T;H s (Ω)) + E t L(,T;H s (Ω)) + E tt L(,T;H s (Ω)) + E ttt L(,T;H s (Ω)) + E tttt L(,T;H s (Ω)) C, H Hs (Ω) + H t L (,T;H s (Ω)) + H t L(,T;H s (Ω)) + H tt L(,T;H s (Ω)) + H ttt L(,T;H s (Ω)) + H tttt L(,T;H s (Ω)) C. We present te main teoretical result in te following teorem. Teorem.. Let (E,H) be te solution of te problem () (4), wit te regularity given by () (), (E n,hn ) be te solution of te discrete sceme (7) (8) and starting approximations (9) (6), wit te finite element polynomial degree

6 56 C. YAO, Y. LIN, C. WANG, AND Y. KOU s k. Under te requirement tat C, wit C a fixed constant, we ave (4) E n E n + H n H n C( + s ), were C is a positive constant, independent of n, te maximum mes size and time step.. Te convergence analysis.. Higer Raviart-Tomas-Nédelec elements. In tis section, we provide an error estimate for te linearized BDF Maxwell FEM sceme (7) (8). As a first step, te starting error estimates are needed. Lemma.. Suppose tat (E,H) is te solution of problem () (4) satisfying () (), wit te finite element polynomial degree s k. Ten te following error estimates of equations (9) (6), for m =,,, are valid: (5) max m Em E m + max m Hm H m C( + s ), were C is a positive constant, independent of n, and. Proof. Te constructed solution (Ẽ, H ) and its interpolation (Π Ẽ,P H ) satisfy te numerical sceme () () up to an O( + s ) truncation error: (6) (7) were (8) (9) () () () ǫ( ΠẼ Π E,ξ u ) (P H, ξ u )+(σ(x, Π E )Π E,ξ u ) = (A,ξ u ), ξ u V, µ( P H P H,ξ φ )+( Π Ẽ,ξ φ ) = (P,ξ φ ), ξ φ W, (A,ξ u ) = (A,ξ u ) (A, ξ u )+(A,ξ u ), ΠẼ Ẽ A = ǫ( + Ẽ E E t )+ H H, A = P H H, A = σ(x, Π E )Π E σ(x, E )E +σ(x, E )E σ(x, E )E, P = µ( P H H + Ẽ E. + H H H t )+ Π Ẽ Ẽ Based on te interpolation error estimates in Lemmas.-. and te Taylor expansion, it is easy to see tat () A C( + s ), (A, ξ u ) =, P C( + s ). For te estimate of A, wit te smoot properties of σ(x,s), te regularity of te exact solution E and te interpolation error estimates, we ave (4) σ(x, E )E σ(x, E )E = σ(x, E )E σ(x, E )E +σ(x, E )E σ(x, E )E σ(x, E )(E E ) + (σ(x, E ) σ(x, E ))E σ(x, E ) L E E +C E E E L E E C,

7 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION 57 and (5) σ(x, Π E )Π E σ(x, E )E = σ(x, Π E )Π E σ(x, Π E )E +σ(x, Π E )E σ(x, E )E σ(x, Π E )(Π E E ) + (σ(x, Π E ) σ(x, E ))E σ(x, Π E ) L Π E E +C Π E E E L C Π E E C s. Ten we arrive at (6) A C( + s ). By denoting A = A +A, we ave A C( + s ), P C( + s ). Define (7) ẽ = Π Ẽ Ẽ, η = P H H. Subtracting (6) (7) from () () yields (8) (9) ǫ(ẽ,ξ u) ( η, ξ u )+(σ(x, Π E )Π E σ(x, E )E,ξ u) = (A,ξ u ), ξ u V, µ( η,ξ φ)+( ẽ,ξ φ ) = (P,ξ φ ), ξ φ W. Taking ξ u = ẽ in (8) and ξ φ = η in (9), summing equations (8) and (9) up, we obtain (4) ǫ(ẽ,ẽ )+µ( η, η ) = (A,ẽ )+(P, η ) (σ(x, Π E )Π E σ(x, E )E,ẽ ). Multiplying equation(4) by on bot sides, employing Young inequality and (5), we ave (4) ǫ ẽ +µ η A ẽ + P η +C Π E E ẽ A 4a +a ẽ + P 4a +a η + C s E 4a +a ẽ. Moreover, by taking ǫ > a +a,µ > a, we see tat (4) ẽ + η C(4 + s ). Forn=, te exactsolution(e,h )anditsinterpolation(π E,P H )satisfy te numerical sceme () (4) up to an O( + s ) truncation error: (4) (44) ǫ( Π E Π E,ξ u ) ( P H +P H, ξ u )+ (σ(x, ΠẼ )Π Ẽ +σ(x, Π E )Π E,ξ u ) = (B,ξ u ), µ( P H P H,ξ φ )+( Π E +Π E,ξ φ ) = (M,ξ φ ),

8 58 C. YAO, Y. LIN, C. WANG, AND Y. KOU were (45) (46) (47) (48) (B,ξ u ) = (B,ξ u ) (B, ξ u )+(B,ξ u ), B = ǫ( Π E E B = P H H, + E E E t )+ H H +H, B = (σ(x, ΠẼ )Π Ẽ +σ(x, Π E )Π E σ(x, Ẽ )Ẽ σ(x, E )E )+ (σ(x, Ẽ )Ẽ +σ(x, E )E ) σ(x, E )E, (49) M = µ( P H H + E +E + H H E. H t )+ Π E E Similarly, te following truncation error estimates are available, based on Lemmas.,.., as well as te Taylor expansion: (5) B C( + s ), (B, ξ u ) =, M C( + s ). Te estimate of B depends on te properties of σ(x,s), te regularity assumption of te exact solution E, and te interpolation error estimates; te details are skipped for te sake of brevity: (5) B C( + s ). Terefore, by defining B = B +B, we ave B C( + s ), M C( + s ). Subsequently, we denote (5) e = Π E E, η = P H H. Subtracting (4) (44) from () (4) results in (5) (54) ǫ( e,ξ u) ( η, ξ u)+ (σ(x, Π Ẽ )Π Ẽ +σ(x, Π E )Π E σ(x, Ẽ )Ẽ σ(x, Π E )E,ξ u) = (B,ξ u ), ξ u V, µ( η,ξ φ)+( e,ξ φ) = (M,ξ φ ), ξ φ W. Taking ξ u = e in (5) and ξ φ = η in (54), summing equations (5) and (54), we arrive at (55) ǫ( e,e )+µ( η,η ) = (B,e )+(M,η ) (σ(x, Π Ẽ )Π Ẽ +σ(x, Π E )Π E σ(x, Ẽ )Ẽ σ(x, Π E )E,e ).

9 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION 59 Te last four terms of (55) could be rewritten as (56) (σ(x, ΠẼ )Π Ẽ +σ(x, Π E )Π E σ(x, Ẽ )Ẽ σ(x, Π E )E ) = (σ(x, ΠẼ )Π Ẽ σ(x, Ẽ )ΠẼ +σ(x, Ẽ )ΠẼ σ(x, Ẽ )Ẽ )+ (σ(x, Π E )(Π E E )+(σ(x, Π E ) σ(x, E ))E ) = ((σ(x, Π Ẽ ) σ(x, Ẽ ))Π Ẽ +σ(x, Ẽ )(Π Ẽ Ẽ )) + (σ(x, Π E )(Π E E )+(σ(x, Π E ) σ(x, E ))E ), so tat te following bound could be obtained: (57) (σ(x, ΠẼ )Π Ẽ +σ(x, Π E )Π E σ(x, Ẽ )Ẽ σ(x, Π E )E ) Π Ẽ Ẽ Π Ẽ L + σ(x, Ẽ ) L Π Ẽ Ẽ + σ(x, Π E ) L Π E E +C Π E E E L C Π Ẽ Ẽ +C Π E E C ẽ +C Π E E. Multiplying (55) by on bot sides and employing Young inequality, we get (58) ǫ e +µ η B e + M η + ẽ e + Π E E e B 4b +b e + M 4b +b η + C ẽ 4b +b e + s E 4b 4 +b 4 e. In addition, by taking b,b,b,b 4 > suc tat b +b +b 4 < ǫ,µ > b, we ave (59) e + η C(6 + s ). Te estimates for n = could be carried out in a similar way. Te following result could be derived, and te details are left to interested readers: (6) e + η C(6 + s ). Tis finises te proof of Lemma.. In addition, te truncation error estimates for n are given below. Proposition.. For te exact solution (E,H), its interpolation (Π E,P H) satisfy te numerical sceme (7) (8) up to an O( + s ) truncation error, in a

10 5 C. YAO, Y. LIN, C. WANG, AND Y. KOU weak form: (6) (6) ǫ(d Π E n,ξ u ) (P H n, ξ u )+(σ(x, Π E n )Π E n σ(x, Π E n )Π E n +σ(x, Π E n )Π E n,ξ u ) = (g n,ξ u), ξ u V, µ(d P H n,ξ φ )+( Π E n,ξ φ ) = (g n,ξ φ ), ξ φ W, wit g n, g n C( + s ). Proof. Obviously, te interpolation (Π E n,p H n ) satisfy te numerical sceme (6) (6) wit te truncated error formula: (6) (g n,ξ u ) = (R n,ξ u ) (R n, ξ u )+(R n,ξ u ), wit (64) R n = ǫ(d Π E n D E n +D E n E n t ), (65) R n = P H n H n, (66) R n = σ(x, Π E n )Π E n σ(x, Π E n )Π E n (67) (68) +σ(x, Π E n )Π E n σ(x, E n )E n, g n = µ(d P H n D H n +D H n H n t)+ Π E n E n. Te interpolation error estimates in Lemmas. and. indicate tat (69) R n C( + s ), (R n, ξ u ) =, g n C( + s ). In addition, te estimate of R n depends on te smoot properties of σ(x,s), te regularity of te exact solution E and te interpolation error estimates. We begin wit a rewritten form: R n := S +S, wit (7) S = σ(x, Π E n )Π E n σ(x, Π E n )Π E n +σ(x, Π E n )Π E n σ(x, Π E n )Π E n, S = σ(x, Π E n )Π E n σ(x, E n )E n. An application of Taylor formula in te tird order expansion implies tat (7) S C. For te term S, te interpolation error estimates are applied: (7) S = σ(x, Π E n )Π E n σ(x, E n )E n σ(x, Π E n )(Π E n E n ) + (σ(x, Π E n ) σ(x, E n ))E n σ(x, Π E n ) L Π E n E n +C Π E n E n E n L C Π E n E n C s. A combination of (7) and (7) leads to (7) R n C( + s ). Terefore, by defining g n = Rn +Rn, we complete te proof of Proposition.. Next, we present te telescope formula in [] for te tird order BDF temporal discretization operator D in te following lemma. Lemma.. Wit te definition of te BDF temporal discrete operator D in (8), tere exists c i, i =,,, c, suc tat (74) (D u n,u n u n ) = c u n c u n + c u n +c u n c u n +c u n + c 4 u n +c 5 u n +c 6 u n c 4u n +c 5 u n +c 6 u n + c 7 u n +c 8 u n +c 9 u n +c u n.

11 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION 5 Now we proceed into te proof of Teorem.. Proof. Te following numerical error functions are defined: (75) e n = Π E n E n, ηn = P H n H n. In turn, subtracting (7) (8) from te consistency estimate (6) (6) yields (76) (77) ǫ(d e n,ξ u ) (η n, ξ u ) = (σ(x, Π E n )Π E n σ(x, E n )E n,ξ u ) +(σ(x, Π E n )Π E n σ(x, E n )E n,ξ u ) (σ(x, Π E n )Π E n σ(x, E n )E n,ξ u )+(g,ξ n u ) := B (ξ u )+B (ξ u )+B (ξ u )+(g n,ξ u), µ(d η n,ξ φ )+( e n,ξ φ ) = (g n,ξ φ). Taking ξ u = e n e n +λ (e n e n ) in (76) and ξ φ = η n η n +λ (η n η n ) in (77), wit certain value λ >, we ave (78) ǫ(d e n,e n e n )+µ(d η n,η n η n ) +ǫλ (D e n,e n e n )+µλ (D η n,η n η n ) = B (ξ u )+B (ξ u )+B (ξ u )+(g n,ξ u)+(g n,ξ φ) +(λ +) ( ( e n,η n ) (η n, e n ) ). Te first two terms in (78) could be analyzed wit te elp of identity (74). Te two additional terms associated wit te temporal difference stencil could be estimates as follows: (D e n,e n e n ) = ( (79) 6 en e n 7 6 (en e n,e n e n ) + ) (en e n,e n e n ) ( en e n 7 en e n ) 6 en e n, (D η n,η n η n ) ( (8) ηn η n 7 ηn η n ) 6 ηn η n, in wic te Caucy inequality as been repeatedly applied. Te last two terms in (78) could be rewritten as (8) ( e n,η n ) (η n, e n ) = ( e n,η n η n ) Te first part in (8) could be bounded as (8) (λ +)( e n,η n η n ) +(η n, (e n e n )). µλ η n η n +C () µ,λ e n, wit C () µ,λ = (λ+) 4µλ. Furtermore, an application of te inverse inequality (8) e Ĉ e,

12 5 C. YAO, Y. LIN, C. WANG, AND Y. KOU implies tat (84) C () µ,λ e n C() µ,λĉ e n C() µ,λ en, wit C () µ,λ = ĈC C() µ,λ, under te requirement tat C. Going back (8), we get (85) (λ +)( e n,η n η n ) µλ η n η n +C() µ,λ en. Te second part in (8) could be similarly analyzed: (86) (λ +)(η n, (e n e n )) α (e n e n ) + 4 (λ +) α η n αĉ e n e n + 4 (λ +) α η n, αĉc e n e n + 4 (λ +) α η n, for any α >. Meanwile, we could coose α wit αĉc = ǫλ, so tat te above inequality becomes (87) (λ +)(η n, (e n e n )) ǫλ e n e n +C() ǫ,λ ηn, wit C () ǫ,λ = 4 (λ + ) α. Consequently, a combination of (8), (85) and (87) leads to (88) (λ +) ( ( e n,η n ) (η n, e n ) ) µλ η n η n + ǫλ e n e n +C () µ,λ en +C () ǫ,λ ηn, under te condition C. Te rest work is focused on te nonlinear error estimate on te rigt and side of (78). We note an L bound for te exact solution and its interpolation (89) E k L C, Π E k L C, in wic te second inequality comes from te following estimate (9) E k Π E k L C s+ ln. An a-priori L assumption up to time step t k,k n We also assume a-priori tat te numerical error function for E as an L bound at time steps t k : (9) e k L, k n, so tat an L bound for te numerical solution E k is available (9) E k L = Π E k e k L = Π E k L + e k L C + := C. Tis assumption will be recovered in later analysis.

13 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION 5 For te nonlinear error term, we begin wit te following decomposition for k = n : (9) σ(x, Π E n )Π E n σ(x, E n )E n = σ(x, Π E n )Π E n σ(x, E n )Π E n +σ(x, E n )Π E n σ(x, E n = (σ(x, Π E n ) σ(x, E n +σ(x, E n )(Π E n E n ) = (σ(x, Π E n ) σ(x, E n )E n ))Π E n ))Π E n +σ(x, E n )e n. Moreover, an application of intermediate value teorem sows tat (94) σ(x, Π E n ) σ(x, E n ) = σ (x,ξ n )e n, wit ξ n between Π E n and E n. On te oter and, te following estimates are available (95) Π E n L C, E n L C, wic coms from te regularity estimate (89) and te a-priori assumption (9), respectively. Ten we get (96) As a consequence, we denote (97) ξ n L Π E n L + E n L C := C + C. C := Tis in turn implies tat (98) max σ (x,θ), so tat σ (x,ξ n ) L C. x Ω, C θ C (σ(x, Π E n ) σ(x, E n ))Π E n = σ (x,ξ n )e n Π E n σ (x,ξ n ) L Π E n L e n C C e n, wit te Hölder inequality applied at te second step. Te analysis for te second term on te rigt and side of (9) is more straigtforward. By te L bound (95) for E n L, we denote (99) C := max σ(x,θ), so tat σ(x, E n ) L C. x Ω, C θ C An application of Hölder inequality sows tat () σ(x, E n )e n σ(x, E n ) L e n C e n. Terefore, a combination of (9), (98) and () yields () σ(x, Π E n )Π E n σ(x, E n )E n C 4 e n, were C 4 = C C + C. As a direct consequence, we obtain te nonlinear error estimate () B (ξ u ) C 4 e n ξ u C C 4 ( e n + e n + e n ), wit te Caucy inequality repeatedly applied. Te following estimates could be similarly derived: () B (ξ u ) C 4 e n ξ u C C 4 ( e n + e n + e n ),

14 54 C. YAO, Y. LIN, C. WANG, AND Y. KOU (4) B (ξ u ) C 4 e n ξ u C C 4 ( e n + en + en ). Te truncation error terms could be bounded by an application of Caucy inequality: (5) (g n,ξ u ) g n ξ u gn +d 5( e n + en ), (6) (g n,ξ φ ) g n ξ φ (7) gn +d 5( η n + ηn ). A substitution of te above estimates into (78) gives ǫ(d e n,e n e n )+µ(d η n,η n η n ) ( +ǫλ 4 en e n 7 en e n 6 en e n ( +µλ 4 ηn η n 7 ηn η n ) 6 ηn η n (C C 4 +d 5 )( e n + en + en + en ) +C () µ,λ en +(C () ǫ,λ +d 5) η n +d 5 η n + ( gn + g n ). By using te telescope formula (74) for D, summing in (7), and applying te discrete Gronwall inequality, we ave ) (8) e n + ηn C 5 ( 6 + s ), wit C 5 dependent on C, independent on and. Recovery of te a-priori bound (9). Wit te elp of te L error estimate (8) and an application of inverse inequality, te following inequality is available, for d : (9) e n L C en d C 6 ( + s ) d C, under a trivial requirement = o( d 6). Tis requirement is muc less severe tan teoneimposed forte linearmaxwellpart, C, associatedwit te stability analysis for te rd order BDF temporal stencil. Ten we ave finised te proof of Teorem.. Remark.. As can be seen in te teoretical derivation, te time step constraint, C, is needed to justify te stability estimate for te rd order BDF temporal stencil. Tis teoretical difficulty comes from te subtle fact tat, te rd order BDF sceme is not A-stable in te classical standard; in particular, te stability domain of te rd order BDF sceme does not contain any part on te purely imaginary axis in te complex plane. On te oter and, it is well-known tat all te eigenvalues associated wit te linear part of Maxwell operator are purely imaginary. Terefore, a direct stability analysis for te rd order BDF sceme is not available for a yperbolic equation; in comparison, suc an analysis could be carefully derived for te parabolic equations, as reported in [].

15 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION 55 Instead of a tird order accurate sceme, if one uses te second order BDF temporal stencil: () () (ǫ En En σ(x, E n (µ Hn Hn + En )E n,ξ u ) (H n, ξ u)+(σ(x, E n )E n,ξ u ) =, ξ u V, n, + Hn,ξ φ )+( E n,ξ φ ) =, ξ φ W, te stability and convergence analyses could be carried out wit a muc milder time step requirement, = O(). Tis improvement is based on te fact tat, te nd order BDF sceme is A-stable, and its stability domain contains te wole part of te purely imaginary axis in te complex plane. In addition to te BDF approximation ()-(), tere ave been some oter second order accurate temporal scemes, suc as Leap-frog (Teorem. [7]) and Crank-Nicolson (see te related reference []), so tat te stability estimate is available for a = O() requirement. However, a teoretical justification of te stability and convergence analyses for any tird order accurate temporal sceme applied to nonlinear Maxwell s equations is still an open problem, and tis article provided te first suc result, under a severe time step constraint, C. On te oter and, suc a severe time step constraint is only a teoretical issue; extensive numerical results ave sown tat, te stability and convergence are well preserved wit a more relaxed constraint, = O(). Tere may be oter alternative tird order accurate temporal scemes for te nonlinear Maxwell s equations, so tat te stability and convergence analyses could be teoretically justified under te milder time step requirement, = O(). Tis investigation will be undertaken in te autors future works... Te lowest Raviart-Tomas-Nédelec element. From te above subsection we observe tat te convergence order estimate for g n and g n in (69)-(7) as played a crucial role to recover te a-priori bound (9). In more details, its spatial accuracy as to be stronger tan O( d ), as demonstrated in (9). In order to improve te convergence order for te lowest Raviart-Tomas-Nédelec element, we ave to employ te super-convergence analysis for a uniform mes; see [9] for te related teoretical tools. Now we consider te lowest Raviart-Tomas-Nédelec element space in tree dimension V = {φ H(curl,Ω),φ K Q,, Q,, Q,,, K T }, W = {ψ [L (Ω)],ψ K Q,, Q,, Q,,, K T }. Te following results are needed in te later analysis; te detailed proofs could be found in [8]. Lemma.. For any ψ V,φ W, denote Π and P as te interpolation operator on V and W, respectively, we ave () () (E Π E,ψ ) = O( ) E ψ, ( (E Π E),φ ) = O( ) E φ.

16 56 C. YAO, Y. LIN, C. WANG, AND Y. KOU Tere exists a post-processing operator Π w Q,,(K),P v Q,,(K) [8, 9], suc tat (4) (5) (6) (i) Π w w C w, P v v C v, w [H (Ω)],v H (Ω), (ii) Π w C w, P v C v, w V,v W, (iii) Π w = Π Π w, P V = P P v, w V,v W, for te adjoin element K = K i,i =,,,4. Using tese postprocessing operators, we can acieve te following global superconvergence for all tree dispersive media: Teorem.. Assume te partition T of Ω is uniform, Π and P are te interpolation on V andw, respectively. If E C 4 (,T;[H (Ω)] ), H C 4 (,T;[H (Ω)] ), for te lowest Raviart-Tomas-Nédelec element space, tere exists te following super-convergence estimate under te condition tat C : (7) (8) max n N E Π En C( + ), max n N H P Hn C( + ), in wic C only depends on te exact solution, independent on and. Proof. From (4)-(6), we ave (9) E Π E n = Π (E n Π E n )+(Π E n E n ) C E n Π E n + Π En E n. Similarly, we ave () H PH n = P(H n P H n )+(PH n H n ) C H n P H n + P Hn H n. From (6) (68) and Lemma., we know tat te following super-convergence estimate is valid () g n, g n ( + ), wit C only dependent on E C 4 (,T;[H (Ω)] ) + H C 4 (,T;[H (Ω)] ). Tis in turn leads to te super-convergentl errorestimate for e n,η n, in a similar way as (8): () e n + ηn C 5 ( 6 + s ), under te a-priori L assumption (9). As a result, suc an assumption could be similarly recovered as (9): () e n L C en d C 5 ( + s ) d Tis finises te argument for te a-priori bound (9). Finally, wit te elp of (4) and (5), we obtain C. (4) E n Π E n C E n, H n P H n C H n. Te proof of Teorem. is completed.

17 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION 57 Error on te mes n+ x n+, n=,,...4 Error on te mes n+ x n+ at iteration step nx for E mes 4x4 mes 8x8 mes 6x6 mes x Error on te mes n+ x n+, n=,, Error on te mes n+ x n+ at iteration step nx for H mes 4x4 mes 8x8 mes 6x6 mes x Iteration step,, Iteration step,,...5 Figure. Error curve on te mes m+ m+, after N = : : 5 time steps. 4. Numerical Results In tis section, we provide some numerical examples in te TE case to confirm our teoretical analysis, wit E = [E,E,] and H = [,,H ]. For convenience, we still denote E = [E,E ] and H = H. Te computations are performed using te Matlab code. Te angular frequency is denoted as ω, and te number of time step is given by a sequence: N = : : 5. We use te lowest Raviart- Tomas-Nédelec element wose basic functions can be found in te section. of [7]. In tese numerical examples, we observe tat, numerical results ave sown tat, te stability and convergence are well preserved wit a more relaxed constraint for te time step, = O(). Te severe time step constraint, C, as appeared in Teorems.,., is only a teoretical issue. Define erre = E n E n, errh = H n H n, SerrE = E n Π E n, SerrH = H n Π H n. Example : Te nonlinear conductivity is fixed as σ( E ) = E E 4. For ω =, te exact solution (E,H) is formulated as E = [e t cos(ωπx)sin(ωπy), e t sin(ωπx)cos(ωπy)], H = ωπe t cos(ωπx)cos(ωπy). Te error curves for E and H on te mes m+ m+, wit m =,,,4, after N = : : 5 time steps are provided in Figure. Te convergence curve is demonstrated in Figure and te numerical solutions are presented in Figure. Example : In te case of rectangular mes, we consider te nonlinear conductivity in te form of σ( E ) = E α,α =.,.5,.6,.8, wit te exact solution (E,H) given by E = [(y )ye y e t,(x )xe x e t ], H = e t (e x ( x +x )+e y (y y +)). Figure 4 sows te convergence curve after time steps wit = 5, and σ( E ) = E α,α =.,.5,.6,.8. Te numerical solution of E and H on te rectangular domain is displayed in Figure 5.

18 58 C. YAO, Y. LIN, C. WANG, AND Y. KOU Convergence Curve on te mes n+ x n+ at iteration step nx for E timestep timestep timestep 4 timestep 5 timestep.. Convergence Curve on te mes n+ x n+ at iteration step nx for H timestep timestep timestep 4 timestep 5 timestep L norm Error L norm Error Mes.9 4 Mes Figure. Convergence after N = : : 5 time steps, wit = Figure. Numerical solution for E and H wit σ( E ) = E E 4. error wit different a for E Á=. Á=.5 Á=.6 Á=.8 error wit different a for H Á=. Á=.5 Á=.6 Á=.8 L norm Error L norm Error 4 4 Mes 4 Mes Figure 4. Convergence curve after time steps wit = 5, and σ( E ) = E α, α =.,.5,.6,.8. Example : In te of an L-sape domain Ω = [,] \[.5,], we consider te nonlinear conductivity in te form of σ( E ) = E α, wit te initial value E (x,y,) = [sin(ωπy),sin(ωπx)], H (x,y,) = ( πω µ )(cos(ωπx) cos(ωπy)),

19 THIRD ORDER BDF SCHEME FOR NONLINEAR MAXWELL S EQUATION Figure 5. Numerical solution for E and H wit σ( E ) = E α,wit α = Figure 6. Numerical solution of E and H on te L-type domain and = e Figure 7. Te figure of σ( E n ) = En α, wit α =.,.5,.6,.8, respectively, at te center of every edge of te element. and te rigt-and-side term F(x,y,t) = (f,f ), f = ǫe t sin(ωπy)+((e t sin(ωπy)) +(e t sin(ωπx)) ) α e t sin(ωπy) 4π ω µ e t sin(ωπy), f = ǫe t sin(ωπx)+((e t sin(ωπy)) +(e t sin(ωπx)) ) α e t sin(ωπx) + 4π ω µ e t sin(ωπx).

20 5 C. YAO, Y. LIN, C. WANG, AND Y. KOU Figure 6 presents te numerical solution of E n and Hn on te L-sape domain. Meanwile, te nonlinear conductivity functions σ( E n ) = En α, wit α =.,.5,.6,.8, are displayed in Figure 7, at te center of every edge of te element, respectively. 5. Conclusions In tis article, we propose a tird order linearized BDF FEM sceme for te nonlinear Maxwell s equations, wit a purely explicit extrapolation applied to te nonlinear terms. Suc an explicit treatment as greatly improved te numerical efficiency; only one constant-coefficient linear system needs to be solved at eac time step. A teoretical analysis for te rd order BDF sceme to te Maxwell s equation turns out to be igly callenging, due to te yperbolic nature of te equation. We make use of a telescope formula for te rd order BDF stencil, combined wit oterrelatedanalytictools, sotat alocalin time stabilitycould be derivedunder a condition for te time step as C, wit C a fixed constant. In addition, te linearized stability analysis for te numerical error function as to be performed, wic in turn yields te full order L error estimate via an L a-priori assumption at te previoustime steps. As a result, an optimal rate L convergenceanalysis and error estimate becomes available. Numerical experiments are investigated, wic confirm te teoretical analysis. Acknowledgments C. Yao is supported by NSFC (NO. 4796, 5789, 84), Y. Lin is supported by Hong Kong Researc Council GRF Grants B-Q56D and 8-ZDA, and C. Wang is supported by NSF DMS References [] A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyedral domains, Numer. Mat.,, 9-65, 5. [] P. Ciarlet and J. Zou, Fully discrete finite element approaces for time-dependent Maxwell s equations, Numer. Mat., 8 (), 9-9, 999. [] T. Cen, T. Kang, G. Lu and L. Wu, A (T,ψ) ψ e decoupled sceme for a time-dependent multiply-connected eddy current problem, Mat. Met. Appl. Sci., 7, 4-59, 4. [4] S. Durand and M. Slodi cka, Convergence of te mixed finite element metod for Maxwell s equations wit nonlinear conductivity, Mat. Met. Appl. Sci.,5, ,. [5] S. Durand and M. Slodička, Fully discrete finite element metod for Maxwell s equations wit nonlinear conductivity, IMA J. Numer. Anal.,, 7-7,. [6] M. V. Ferreira and C. Buriol, Ortogonal decomposition and asymptotic beavior for nonlinear Maxwell s equations, J. Mat. Anal. Appl., 46, 9-45, 5. [7] H. Gao, Unconditional optimal error estimates of BDF-Galerkin FEMs for nonlinear termistor equations, J. Sci. Comput, 5. [8] H. Gao, Optimal Error Analysis of Galerkin FEMs for nonlinear Joule eating equations, J. Sci. Comput., 58, , 4. [9] Y. Q. Huang, J. C. Li, C. Wu, and W. Yang, Superconvergence analysis for linear tetraedral edge elements, J. Sci. Comput., 6, -45, 5. [] Y. Q. Huang, J. C. Li and W. Yang, Solving metamaterial Maxwell s equations via a vector wave integro-differential equation, Comput. Mat. Appl., 6, ,. [] Y. Q. Huang, J. C. Li and Q. Lin, Superconvergence analysis for time-dependent Maxwell s equations in metamaterials, Numer. Metods Partial Diff. Equ., 8, ,. [] Y. Q. Huang, J. C. Li and C. Wu, Averaging for superconvergence: verification and application of D edge elements to Maxwell s equations in metamaterials, Comput. Met. Appl. Mec. Engrg. 55, -,. [] X. Jiang and W.Y. Zeng, An efficient eddy current model for nonlinear Maxwell equation wit laminated conductors, SIAM J. Appl. Mat. 7, -4,.

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