Semi-discrete finite element approximation applied to Maxwell s equations in nonlinear media

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1 Semi-discrete finite element approximation applied to Maxwell s equations in nonlinear media Lutz Angermann arxiv:9.365v math.na Jan 9 January 9, 9 In this paper the semi-discrete finite element approximation of initial boundary value problems for Maxwell s equations in nonliear media of Kerr-type is investigated. For the case of Nédélec elements from the first family, a priori error estimates are established for the approximation. Keywords: Semi-discrete finite element method, nonlinear Maxwell s equations, error estimate MSC : 35Q6, 65M6, 65M5 Introduction In this paper we investigate the semi-discrete conforming finite element approximation to the solution of Maxwell s equations for nonliear media of Kerr-type. As a concrete example, we consider the meanwhile classical) Nédélec elements from the so-called first family. To the best knowlege of the author, the nonlinear situation is not yet well investigated, most works dealing with nonlinear effects are computational or experimental see, e.g., AY8). Here we derive energy stability) estimates for the weakly formulated problem and error estimates for the semi-discretized problem. Let Q T :=, T ), where R 3 is a simply connected domain with a sufficiently smooth boundary and T > is the length of the time interval under consideration. Let D, B, E, H : Q T R 3 represent the displacement field, the magnetic induction, the electric and magnetic field intensities, respectively. The time-dependent Maxwell s equations in a nonlinear medium can be written in the form t D H = in Q T, ) t B + E = in Q T, ) Institut für Mathematik, Technische Universität Clausthal, Erzstraße, D Clausthal-Zellerfeld, Germany, lutz.angermann@tu-clausthal.de

2 where the following constitutive relations hold: B := µ H, D := ε E + PE). 3) Here ε > and µ > are the vacuum permittivity and the permeability, respectively. Often the constitutive relation for the polarization P = PE) is approximated by a truncated Taylor series Boy3. In the case of an isotropic material, it takes the form ) PE) := ε χ ) E + χ 3) E E, where χ j) : R are the media susceptibility coefficients, j =, 3. Then from 3) we obtain the representation D = ε ε s E with ε s = ε s E) := + χ ) + χ 3) E, and it follows by a simple calculation that t D = ε χ 3) E t E)E + ε ε s t E = ε ε s I + ε m ) t E with ε m = ε m E) := χ 3) EE, where I denotes the identy in R d. Setting the system ) 3) can be written as εe) := ε ε s E)I + ε m E)), εe) t E H = in Q T, 4) µ t H + E = in Q T. 5) Next we state a simple result which in particular implies that the matrix εe) is regular for all electric field intensities E under consideration. Lemma.. Let χ ), χ 3) a.e. in. Then the symmetric matrix εψ) is uniformly positive definite a.e. for any Ψ R 3. Proof. For all Φ R 3, it holds that ε Φ εψ)φ = + χ ) + χ 3) Ψ ) Φ + χ 3) Ψ Φ Φ. As in PNTB9, we denote the inverse by CE) := εe) ). By means of the Sherman-Morrison formula see, e.g., GvL96), the matrix CE) can be given explicitely: CE) = ε C m E) 6)

3 with C m E) := ε s E)I + ε m E)) = ε s E) = I ε s E) I + χ ) + 3χ 3) E ε me) ) ε s E) + χ 3) E ε me) ). Therefore, if the formula 6) holds, the system 4) 5) takes the form ε t E C m E) H = in Q T, 7) µ t H + E = in Q T. 8) This is an appropriate formulation for the development of time-discrete numerical algorithms, see, e.g., PNTB9. A perfect conducting boundary condition on is assumed so that ν E = on, T ), 9) where ν denotes the outward unit normal on. In addition, initial conditions have to be specified so that E, x) = E x) and H, x) = H x) for all x, ) where E : R 3 and H : R 3 are given functions, and H satisfies µ H ) = in, H ν = on. ) The divergence-free condition in ) together with 5) implies that Notation µ H) = in Q T, ) For a real number p,, the symbol L p ) denotes the usual Lebesgue spaces equipped with the norm L p ). The analogous spaces of vector fields u : R 3 are denoted by L p ) := L p ) 3 with the norm L p ). In what follows we have to deal with weighted function spaces. Given a weight ω : R, where the values of ω are positive a.e. on, we define a weighted inner product and a weighted norm by u, v) ω := ω u v dx and u ω := u L ω ) := u, u) ω. 3) The space L ω) consists of vector fields u : R 3 with Lebesgue-measurable components and such that u ω <. In the case ω =, the subscript is omitted. An elementary property of weighted spaces, which we will apply at different places without special emphasis, is the monotonicity w.r.t. the weight: If ω, ω are two weights such that ω ω a.e. on, then u L ω ) u L ω ) for all u L ω ) L ω ). 3

4 As transient problems are addressed, we will work with functions that depend on time and have values in certain Banach spaces. If u = ut, x) is a vector field of the space variable x and the time variable t, it is suitable to separate these variables in such a way that ut) = ut, ) is considered as a function of t with values in a Banach space, say X, with the norm X. That is, for any t, T ), the mapping x ut, x) is interpreted as a parameter-dependent element ut) of X. In this sense we will write Et) = Et, ), Ht) = Ht, ) and so on. The space C m, T ; X), m N {}, consists of all continuous functions u :, T ) X that have continuous derivatives up to order m on, T ). It is equipped with the norm u C m,t ;X) := m j= sup t,t ) u j) t) X. For the sake of consistency in the notation we will write C, T ; X) := C, T ; X). The space L p, T ; X) with p, ) contains equivalent classes of) strongly measurable functions u :, T ) X such that T ut) p X dt < for the definition of strongly measurable functions we refer to KJF77). L p, T ; X) is defined by The norm on { T } /p u Lp,T ;X) := ut) p X dt. These spaces can be equipped with a weight, too. In particular, we will write { T u L,T ;L ω )) := ut) ω dxdt} /. Finally, all the above definitions can be extended to the standard Sobolov spaces of functions with weak spatial derivatives of maximal order r N in L p ): W r,p ) with norm W r,p ). If p =, we write H r ) := W r, ) and H r ) := W r, ). The space H) is defined as the closure of C ) with respect to the norm H ), where C ) denotes the space of all arbitrarily often differentiable functions with compact support on. It is well knwon that H) is a closed subspace of H ) and consists of elements u such that u = on in the sense of traces AF3. As in the case of the L p -spaces, we shall write W r,p ) := W r,p ) 3 and so on. Furthermore, we need the following Hilbert spaces that are related to the weak) rotation and divergence operators: Hcurl, ) := {u L ) : u L )}, H curl, ) := {u Hcurl, ) : u ν = }, Hdiv, ) := {u L ) : u L )}. 4

5 These spaces are equipped with the norms resp. induced norms) u Hcurl,) := { u + u } /, u Hdiv,) := { u + u } /. We refer to DL76, RT77, GR86 and Cia for details about these spaces. 3 Weak formulations We assume that a solution E, H) C, T ; L,εE) )) C, T ; H curl, )) ) C, T ; L )) C, T ; Hcurl, )) ) of the nonlinear Maxwell s equations 4) 5) exists and is unique. We multiply equation 4) by a test function Ψ L ) and integrate over. Similarly we multiply 5) by a test function Φ Hcurl, ), integrate the result over and integrate by parts the second term. This shows that it is natural to look for a weak solution E, H) C, T ; L,εE) )) C, T ; L )) ) C, T ; L )) C, T ; Hcurl, )) ) of 4) 5) such that εe) t E, Ψ) H, Ψ) = Ψ L ), 4) µ t H, Φ) + E, Φ) = Φ Hcurl, ). 5) Alternatively, the use of test functions Ψ H curl, ) and Φ L ) and the integration by parts in the equation 4) leads to the notion of a weak solution E, H) C, T ; L,εE) )) C, T ; H curl, )) ) C, T ; L )) ) of 4) 5) such that εe) t E, Ψ) H, Ψ) = Ψ H curl, ), 6) µ t H, Φ) + E, Φ) = Φ L ). 7) In both cases, the initial conditions ) have to be satisfied at least in the sense of C, T ; L )). Remark. As a consequence of the embedding as sets) C ) 3 C ) H ) 3 H ) 3 Hdiv, ) L ) and of the fact that C ) is dense in L ) we see that Hdiv, ) is a dense subset of L ) AF3. Therefore the test space in 7) can be reduced to Hdiv, ). Also note that H C, T ; Hdiv, )) due to ). Remark. In the case where µ is not a constant but a highly variable function µ = µx) it is more convenient to use the magnetic flux density B = µh instead of H as a dependent variable MM95. In such a case, the formulation 6) 7) is replaced by where εe) t E, Ψ) µ B, Ψ) = Ψ H curl, ), 8) µ t B, Φ) + E, µ Φ) = Φ Hdiv, ), 9) E, B) C, T ; L,εE) )) C, T ; H curl, )) ) C, T ; L µ ))) C, T ; Hdiv, )) ). 5

6 Next we will formulate a stability result for the problem 4) 5). For this, we extend this problem to the case of a nontrivial right-hand side for a moment. Theorem 3.. Let J e L, T ; L,ε +χ) ) )), J m L, T ; L,µ )) the electric and magnetic current densities, respectively be given and assume that the system εe) t E, Ψ) H, Ψ) = J e, Ψ) Ψ L ), ) µ t H, Φ) + E, Φ) = J m, Φ) Φ Hcurl, ) ) together with the initial conditions ) has a weak solution E, H) C, T ; L,εE) )) C, T ; H curl, )) ) C, T ; L )) ). If W t) := Et) ε +χ ) ) + 3 Et) ε + Ht) χ 3) µ denotes the nonlinear electromagnetic energy at the time t, the following energy law in differential form holds: dw dt = J e, E) J m, H), t, T ). ) Proof. Taking Ψ = E and Φ = H in ) ) and adding the result gives An elementary calculation shows that hence εe) t E, E) + µ t H, H) = J e, E) J m, H) 3) εe) t E = ε + χ ) + χ 3) E )I + χ 3) EE ) t E = ε + χ ) + χ 3) E ) t E + ε χ 3) EE t E) = ε + χ ) + χ 3) E ) t E + ε χ 3) E t E, E εe) t E) = ε + χ ) + χ 3) E )E t E + ε χ 3) E E t E = ε + χ) + χ 3) E ) t E + ε χ 3) E t E = ε + χ) ) t E + 3ε χ3) E t E = ε + χ) ) t E + 3ε 4 χ3) t E 4. Then it follows from equation 3) that dw = d E ε dt dt +χ ) ) + 3 E ε + H χ 3) µ = J e, E) J m, H). Corollary 3.. Under the assumptions of Thm. 3., for all t, T ), the following estimate is valid: W t) W ) + t J e L,T ;L,ε +χ) ) )) + J m L,T ;L,µ )). 6

7 Proof. We estimate the right-hand side of ) by means of the Cauchy-Bunyakovski-Schwarz inequality twice in the integral version and in the finite sum version): J e, E) + J m, H) J e ε +χ) ) E ε +χ ) ) + J m µ H µ { } / } / { E ε+χ )) + H,µ J e ε +χ) ) + J m µ { J e + J ε +χ) ) m µ } / W t). Since dt d W = dw for W t) > the case W t) = is trivial), it follows from ) that W dt d { } / W t) J e + J dt ε +χ) ) m µ. Integrating this inequality w.r.t. t, we get Then W t) W ) + t t W t) W ) + t W ) + t W ) + t J e L,T ;L,ε { } / J e + J ε +χ) ) m µ ds. { } ) / J e + J ε +χ) ) m µ ds J e + J ε +χ) ) m µ ds +χ) ) )) + J m L,T ;L,µ )). Remark 3. Analogous results as in Thm. 3. and Cor. 3. can be obtained for the corresponding non-homogeneous version of 6) 7) and for the subsequent semi-discretizations. 4 Spatial discretization 4. Semi-discretization of the weak formulations Let W h L ), U h Hcurl, ), U h H curl, ), and V h Hdiv, ) be finitedimensional subspaces. The semi-discrete in space) problem for the system 4) 5) consists in determining elements E h, H h ) C, T ; W h ) C, T ; U h ) such that εe h ) t E h, Ψ h ) H h, Ψ h ) = Ψ h W h, 4) µ t H h, Φ h ) + E h, Φ h ) = Φ h U h. 5) For the the equations 6) 7), the semi-discrete problem involves the determination of elements E h, H h ) C, T ; U h ) C, T ; V h ) such that εe h ) t E h, Ψ h ) H h, Ψ h ) = Ψ h U h, 6) 7

8 µ t H h, Φ h ) + E h, Φ h ) = Φ h V h. 7) The initial conditions for both problems read formally as E h, x) = E h x) and H h, x) = H h x) for all x, where the concrete requirements to the particular choice of the discrete initial data E h, H h ) will be seen later Thm. 5.). 4. The choice of the finite element spaces In the rest of the paper we will choose the so-called first family of Nédélec edge elements, usually denoted by D k and R k for k N, for the construction of the concrete finite element spaces for details see Néd8 or Mon3, Ch. 5). That is, given an arbitrary member T h of a family of triangulations of consisting of open tetrahedra K, we set U h := {w Hcurl; ) : w K R k K T h }, V h := {w Hdiv; ) : w K D k K T h }, W h := {w L ) : w K P k K T h }, where P l := P l 3 and P l is the space of scalar real-valued polynomials in three variables of maximal degree l N {}. To deal with the case of the Nédélec formulation 6) 7), we still have to introduce the space U h := U h H curl, ). In the subsequent error analysis, we will make use of some projection operators. For the Lee-Madsen formulation 4) 5), we need projections P h : L ) W h, Π h : Hcurl, ) U h. Let P h be the standard L )-projection operator onto W h, i.e. for given w L ) the image P h w W h is defined by P h w, Ψ h ) = w, Ψ h ), Ψ W h. 8) For this operator the following standard error estimate holds: If w H k ), then w P h w Ch k w H k ). 9) Moreover, since U h W h see the beginnings of the proofs of Mon9, Thm. 3.3 or Mon3, Lemma 5.4), it holds that P h w, Φ h ) = w, Φ h ), Φ U h. 3) Next, for v Hcurl, ) we define Π h v U h by Πh v, Ψ h ) = v, Ψh ) Ψ h U h, 3) Π h v, p h ) = v, p h ) p h S k h, 3) where S k h is defined as S k h := {v H )/R : v K P k K T h }, see Mon9, Subsect. 4.. If v H k+ ) such that v = in and ν v = on, then there exists a constant C > such that v Π h v Ch k v H k+ ) 33) see Mon9, Thm. 4.6). 8

9 5 An error estimate for the semi-discrete problem In this section we formulate and prove the main result. Theorem 5. Semi-discrete error estimate for the Lee-Madsen formulation). Let k N, χ ), χ 3) L ), E L ), H L,µ ) satisfying ), E, H ) C, T ; L ) H curl, )) E C, T ; H k )) ) C, T ; H k+ )) be the weak solution of the system 4) 5), and Eh, H h ) C, T ; Wh ) C, T ; L )) C, T ; U h ) be the finite element solution of the system 4) 5) respectively, where the inclusion is to be understood uniformly w.r.t. the mesh parameter h in the sense that E h C,T ;L )) is bounded by a constant independent of h. Then there exists a constant C > independent of h such that the following error estimate holds: E h T ) ET ) ε + H h T ) HT ) µ C P h E E h ε + Π h H H h µ + h k the detailed structure of the bound is given at the end of the proof). Proof. We set Ψ := Ψ h W h in 4) and Φ := Φ h U h in 5): εe) t E, Ψ h ) H, Ψ h ) = Ψ h W h, µ t H, Φ h ) + E, Φ h ) = Φ h U h. By means of the projection operators P h and Π h defined in 8) and 3) 3), resp., from this we get εe) t E, Ψ h ) Π h H, Ψ h ) = H Π h H), Ψ h ) Ψ h W h, 34) µ t Π h H, Φ h ) + P h E, Φ h ) = µ Π h t H t H, Φ h ) + µ t Π h H Π h t H, Φ h ) 35) + P h E E, Φ h ) Φ h U h. The last term on the right-hand side of 34) vanishes thanks to the properties of Π h, see Mon9, eq..4) and 3). The second term on the right-hand side of 35) can be omitted because of the commutation property t Π h H = Π h t H, which results from the continuity properties of the operator Π h. The last term on the right-hand side vanishes thanks to the property 3) of P h. Therefore 34) 35) simplify to εe) t E, Ψ h ) Π h H, Ψ h ) = Ψ h W h, 36) µ t Π h H, Φ h ) + P h E, Φ h ) = µ Π h t H t H, Φ h ) Φ h U h. 37) Now, subtracting 36) 37) from the system 4) 5) and taking into consideration that µ is constant, we obtain: εe h ) t E h εe) t E, Ψ h ) H h Π h H), Ψ h ) = Ψ h W h, 38) 9

10 µ t H h Π h H), Φ h ) + E h P h E, Φ h ) = µ t H Π h t H, Φ h ) Φ h U h. 39) Now we will deal with the first term of 38), where we have in mind the choice Ψ h = E h P h E in what follows: ε εe h ) t E h εe) t E = ε s E h )I + ε m E h )) t E h ε s E)I + ε m E)) t E) = + χ ) + χ 3) E h )I + ε m E h ) ) t E h + χ ) + χ 3) E )I + ε m E) ) t E = + χ ) ) t E h + χ ) ) t E + χ 3) E h I + ε m E h ) ) t E h χ 3) E I + ε m E) ) t E = + χ ) ) t E h E) + χ 3) E h t E h E t E + χ 3) E h E h t E h EE t E =: δ + δ + δ 3. The treatment of δ is quite obvious. With E h E = Ψ h + P h E E we get δ = + χ ) ) t Ψ h + + χ ) ) t P h E E) =: δ + δ. The term δ is decomposed as follows: δ = χ 3) E h t E h E t E = χ 3) E h E t E h + χ 3) E t E h E) = χ 3) E h + E) E h E) t E h + χ 3) E t Ψ h + χ 3) E t P h E E) =: δ + δ + δ 3. For δ 3, we use the following decomposition: δ 3 = χ 3) E h E h t E h EE t E = χ 3) E h E h EE t E h + χ 3) EE t E h E) = χ 3) E h E)E h t E h + χ 3) EE h E) t E h + χ 3) EE t Ψ h + χ 3) EE t P h E E) =: δ 3 + δ 3 + δ 33 + δ 34. With these decompositions, equation 38) takes the form εe h ) t E h εe) t E, Ψ h ) H h Π h H), Ψ h ) = ε δ + δ + δ 33 Ψ h dx + ε δ + δ + δ 3 + δ 3 + δ 3 + δ 34 Ψ h dx H h Π h H), Ψ h ) =, or, after some rearrangement, ε δ + δ + δ 33 Ψ h dx H h Π h H), Ψ h )

11 Then: Since ε = ε = ε = ε δ + δ + δ 33 Ψ h dx δ + δ + δ 3 + δ 3 + δ 3 + δ 34 Ψ h dx. 4) + χ ) ) t Ψ h Ψ h + χ 3) E t Ψ h Ψ h + χ 3) EE t Ψ h ) Ψh dx + χ ) ) t Ψ h + χ 3) E t Ψ h + 4χ 3) E t Ψ h E Ψ h dx. E t Ψ h = t E Ψ h ) t E ) Ψ h and E t Ψ h = t E Ψ h ) t E Ψ h, it follows that ε = ε From the estimates ε and, analogously, ε we conclude that ε δ + δ + δ 33 Ψ h dx + χ ) ) t Ψ h + χ 3) t E Ψ h ) χ 3) t E ) Ψ h + 4χ 3) t E Ψ h )E Ψ h 4χ 3) t E Ψ h E Ψ h dx = ε + χ ) ) t Ψ h + χ 3) t E Ψ h ) χ 3) t E ) Ψ h + χ 3) t E Ψ h 4χ 3) t E Ψ h E Ψ h dx = ε + χ ) ) t Ψ h + χ 3) t E Ψ h ) + χ 3) t E Ψ h dx ε χ 3) t E ) Ψ h dx ε χ 3) t E Ψ h E Ψ h dx. ε χ 3) t E ) Ψ h dx = ε χ 3) t E E Ψ h dx χ 3) L ) t E C,T ;L )) E C,T ;L )) Ψ h ε χ 3) L ) E C,T ;L )) Ψ h ε χ 3) t E Ψ h E Ψ h dx χ3) L ) E C,T ;L Ψ )) h ε δ + δ + δ 33 Ψ h dx + χ ) ) t Ψ h + χ 3) t E Ψ h ) + χ 3) t E Ψ h dx

12 χ 3) L ) E C,T ;L Ψ )) h ε = t Ψ h ε +χ ) ) + ε t χ 3) E Ψ h + E Ψ h dx χ 3) L ) E C,T ;L )) Ψ h ε. 4) For the right-hand side, we have: ε δ + δ + δ 3 + δ 3 + δ 3 + δ 34 Ψ h dx = ε + χ ) ) t P h E E) Ψ h + χ 3) E h + E) E h E) t E h Ψ h + χ 3) E t P h E E) Ψ h + χ 3) E h E)E h t E h ) Ψh + χ 3) EE h E) t E h ) Ψh + χ 3) EE t P h E E) ) Ψh dx ε + χ ) ) t P h E E) Ψ h + χ 3) E h + E E h E t E h Ψ h + χ 3) E t P h E E) Ψ h + χ 3) E h t E h E h E) Ψ h + χ 3) E h E) t E h E Ψ h + χ 3) E t P h E E)E Ψ h dx ε + χ ) + χ 3) E ) t P h E E) Ψ h + χ 3) E h + E t E h Ψ h + χ 3) E h + E P h E E t E h Ψ h + χ 3) E h t E h Ψ h + χ 3) E h t E h P h E E) Ψ h + χ 3) Ψ h t E h E Ψ h + χ 3) P h E E) t E h E Ψ h + χ 3) E t P h E E)E Ψ h dx ε + χ ) + χ 3) E ) t P h E E) Ψ h + χ 3) E h t E h Ψ h + χ 3) E t E h Ψ h + χ 3) E h P h E E t E h Ψ h + χ 3) E P h E E t E h Ψ h + χ 3) E h t E h Ψ h + χ 3) E h t E h P h E E Ψ h + χ 3) E t E h Ψ h + χ 3) E t E h P h E E Ψ h + χ 3) E t P h E E) Ψ h dx = ε + χ ) + 3χ 3) E ) t P h E E) Ψ h + 3χ 3) E h t E h P h E E Ψ h + 3χ 3) E t E h P h E E Ψ h + 3χ 3) E h t E h Ψ h + 3χ 3) E t E h Ψ h dx = ε + χ ) + 3χ 3) E ) t P h E E) Ψ h + 3χ 3) E h + E ) t E h P h E E Ψ h + 3χ 3) E h + E ) t E h Ψ h dx

13 + χ ) L ) + 3 χ 3) L ) E C,T ;L )) t P h E E) ε Ψ h ε + 3 χ 3) L ) Eh C,T ;L )) + E C,T ;L )) t E h C,T ;L )) P h E E ε Ψ h ε + 3 χ 3) L ) Eh C,T ;L )) + E C,T ;L )) t E h C,T ;L )) Ψ h ε =: C t P h E E) ε Ψ h ε + C P h E E ε Ψ h ε + C 3 Ψ h ε, 4) where the positive constants C, C, C 3 depend on certain norms of χ ), χ 3), E, and E h. Combining the estimates 4) and 4) with 4), we get t Ψ h ε +χ ) ) + ε t χ 3) E Ψ h + E Ψ h dx This finally leads to where χ 3) L ) E C,T ;L Ψ )) h ε H h Π h H), Ψ h ) ε δ + δ + δ 33 Ψ h dx H h Π h H), Ψ h ) = ε δ + δ + δ 3 + δ 3 + δ 3 + δ 34 Ψ h dx C t P h E E) ε Ψ h ε + C P h E E ε Ψ h ε + C 3 Ψ h ε. t Ψ h ε +χ ) ) + ε t H h Π h H), Ψ h ) χ 3) E Ψ h + E Ψ h dx C t P h E E) ε + C P h E E ε Ψ h ε + C 4 Ψ h ε, C 4 := C 3 + χ 3) L ) E C,T ;L )). Now we consider 39) with Φ h = H h Π h H and get t Φ h µ + E h P h E, Φ h ) = µ t H Π h t H, Φ h ) t H Π h t H µ Φ h µ. Adding both inequalities and making use of the commutation property of P h, we arrive at t Ψ h ε +χ ) ) + t Φ h µ + ε t χ 3) E Ψ h + E Ψ h dx C t E P h t E ε + C E P h E ε Ψ h ε + t H Π h t H µ Φ h µ + C 4 Ψ h ε. The projection errors can be estimated by means of 9) and 33), that is, for E, t E H k ) and t H H k+ ), we have that E P h E ε C ε h k E H k ) C ε h k E C,T ;H k )), t E P h t E ε C ε h k t E H k ) C ε h k t E C,T ;H k )), t H Π h t H µ C µ h k t H H k+ ) C µ h k t H C,T ;H k+ )). 3

14 In this way the above estimate can be written as t Ψ h ε +χ ) ) + t Φ h µ + ε t C 5 h k Ψ h ε + Φ h µ + C 4 Ψ h ε. χ 3) E Ψ h + E Ψ h dx Setting we get w h t) := Ψ h t) ε + Φ h t) µ, t Ψ h ε +χ ) ) + t Φ h µ + ε t χ 3) E Ψ h + E Ψ h dx C 5 h k w h t) + C 4 Ψ h ε C 5 h k w h t) + C 4 w ht). Integrating this inequality, we obtain Ψ ht) ε +χ ) ) + Φ ht) µ + ε χ 3) Et) Ψ h t) + Et) Ψ h t) dx Ψ h) ε +χ ) ) + Φ h) µ + ε χ 3) E) Ψ h ) + E) Ψ h ) dx t + C 5 h k w h s) + C 4 whs) ds. 43) By the monotonicity of the weighted norms w.r.t. the weight and the nonnegativity of the integral term on the left-hand side, we see that w ht) Ψ ht) ε +χ ) ) + Φ ht) µ + ε χ 3) Et) Ψ h t) + Et) Ψ h t) dx. 44) On the other hand, we have the estimates Ψ h ) ε +χ ) ) + χ) L ) Ψ h ) ε + χ ) L )w h) 45) and ε χ 3) E) Ψ h ) + E) Ψ h ) dx 3 χ 3) L ) E) L ) Ψ h ) ε 3 χ 3) L ) E) L )w h). 46) 4

15 Combining 44), 45), 46) with 43), we get or, equivalently, w ht) + χ) L )wh) + 3 χ3) L ) E) L )wh) t + C 5 h k w h s) + C 4 whs) ds, w ht) C 6w h) + t C 5 h k w h s) + C 4 whs) ds, 47) where C 6 := + χ ) L ) + 3 χ 3) L ) E) L ). In the paper Daf79, a Gronwall-type lemma Lemma 4.) is specified which gives a bound on the value wt ) provided an inequality like 47) is satisfied: w h T ) C 6 e C 4T w h ) + C 5 h k T e C 4T. From this and the triangle inequality in conjunction with 9) and 33) the statement follows. Indeed, since wt) Ψ h t) ε + Φ h t) µ wt) for all t, T, we get Ψ h T ) ε + Φ h T ) µ C 6 w h ) + C 5 h k T Then e C 4T C 6 Ψ h ) ε + C 6 Φ h ) µ + C 5 h k T e C 4T. E h T ) ET ) ε + H h T ) HT ) µ Ψ h T ) ε + Φ h T ) µ + P h ET ) ET ) ε + Π h HT ) HT ) µ C 6 Ψ h ) ε + C 6 Φ h ) µ + C 5 h k T e C 4T + Ch k ε E C,T ;H k )) + µ H C,T ;H k+ )), which implies the stated estimate. Remark 4. Note that the constant C in this estimate behaves as T e C 4T for large T. 6 Conclusion In this paper we have investigated a semi-discrete conforming finite element approximation to the solution of Maxwell s equations for nonliear media of Kerr-type using Nédélec elements from the first family. We have demonstrated energy stability) estimates for the weakly formulated problem and error estimates for the semi-discretized problem. The results can be extended to other conforming finite element methods provided the corresponding projection operators P h and Π h 8), 3) 3)) admit analogous properties. 5

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