A symplectic discontinuous Galerkin full discretization for stochastic Maxwell equations
aa r X i v : . [ m a t h . NA ] S e p A SYMPLECTIC DISCONTINUOUS GALERKIN FULL DISCRETIZATION FORSTOCHASTIC MAXWELL EQUATIONS
CHUCHU CHENA
BSTRACT . This paper proposes a fully discrete method called the symplectic dG full discretiza-tion for stochastic Maxwell equations driven by additive noises, based on a stochastic symplecticmethod in time and a discontinuous Galerkin (dG) method with the upwind fluxes in space.A priori H k -regularity ( k ∈ { , } ) estimates for the solution of stochastic Maxwell equationsare presented, which have not been reported before to the best of our knowledge. These H k -regularities are vital to make the assumptions of the mean-square convergence analysis on theinitial fields, the noise and the medium coefficients, but not on the solution itself. The conver-gence order of the symplectic dG full discretization is shown to be k / k − /
1. I
NTRODUCTION
Stochastic Maxwell equations are often used to better understand the role of thermodynamicfluctuations presented in the electromagnetic fields, and to get a deeper insight regarding thepropagation of electromagnetic waves in complex media (see e.g. [16]). A mathematicallyrigorous framework on the effects of randomness has been developed in [15]. The numericaltreatment of the three dimensional stochastic Maxwell equations, even in the linear case, is achallenging task, due to the interaction of the large scale and the randomness of the problem.In this paper, we first discretize stochastic Maxwell equations in time via the midpoint scheme,which inherits the stochastic symplecticity of the original continuous problem, and subsequentlyin space based on a dG method combining its attractive features on the treatment of complex ge-ometries and composite media.For the time-dependent stochastic Maxwell equations, there exist some works on the con-struction of full discretizations, for example, multi-symplectic numerical methods (cf. [7, 12]),energy-conserving methods (cf. [13]). On the rigorous error analysis of the numerical approxi-mations, the existing works mainly focus on the temporal semidiscretizations (see [5, 6, 8]). It isshown in [5] that a semi-implicit Euler scheme converges with order 1 / /
2, when ap-plied to stochastic Maxwell equation with multiplicative Itˆo noise. Authors in [6] show that thestochastic symplectic Runge-Kutta semidiscretizations are mean-square convergent with order1 in the additive case. As far as we know, there are few works on the rigorous error analysisof the spatio-temporal full discretizations for the time-dependent stochastic Maxwell equations.
Key words and phrases.
Stochastic Maxwell equations, Symplectic dG full discretization, Mean-squareconvergence.This work is funded by National Natural Science Foundation of China (No. 11871068, No. 12022118, No.11971470 and No. 12031020).
The difficulty lies in the lack of regularity of the solution in H k -norms or even in C k -norms,which depends on the spatial domain, the medium coefficients and the noise, etc. For exam-ple, on a cuboid, the solution of the time-harmonic deterministic Maxwell equations only has H α -regularity for α < E ( t , x ) and H ( t , x ) satisfying the following stochastic Maxwell equations on a cuboid D =( a − , a + ) × ( a − , a + ) × ( a − , a + ) ⊂ R , ε d E − ∇ × H d t = − d W e ( t ) , ( t , x ) ∈ ( , T ] × D , (1.1a) µ d H + ∇ × E d t = − d W m ( t ) , ( t , x ) ∈ ( , T ] × D , (1.1b) ∇ · ( ε E ) = , ∇ · ( µ H ) = , ( t , x ) ∈ ( , T ] × D , (1.1c) n × E = , n · ( µ H ) = , ( t , x ) ∈ ( , T ] × ∂ D , (1.1d) E ( , x ) = E ( x ) , H ( , x ) = H ( x ) , x ∈ D , (1.1e)where T >
0, and n ( x ) denotes the outer unit normal at x ∈ ∂ D . We suppose that the mediumis isotropic, which implies that the permittivity ε and the permeability µ are real-valued scalarfunctions, i.e., ε , µ : D → R . Throughout this paper, we assume the medium coefficients satisfy ε , µ ∈ L ∞ ( D ) , ε , µ ≥ δ for a constant δ > . (1.2)Here W e ( t ) (resp. W m ( t ) ) is a Q e -Wiener (resp. Q m -Wiener) process with respect to a filteredprobability space ( Ω , F , { F t } ≤ t ≤ T , P ) with Q e (resp. Q m ) being a symmetric, positive definiteoperator with finite trace on U = L ( D ) . Moreover, W e ( t ) and W m ( t ) are independent. Thephase flow of (1.1) preserves the stochastic symplecticity (cf. [6]), i.e., if ε , µ are constants, forany t ∈ [ , T ] , ω ( t ) = R D d E ( t ) ∧ d H ( t ) = ω ( ) , P -a.s.The solution theory of (1.1), which is crucial in the mean-square error analysis, is pre-sented in Section 2 with certain assumptions being made on the medium coefficients, the ini-tial fields and the noise. We restrict the Maxwell operator M on the closed subspace V of V : = L ( D ) × L ( D ) , in order to respect to all boundary conditions and divergence proper-ties. These conditions and properties are important to get the L p ( Ω ; C ([ , T ] ; H ( D ) )) -regularity( H -regularity in short) for the solution of (1.1), under the first order regularity and certain com-patibility conditions of the initial data and the noise term; see Proposition 2.1. Furthermore,we can guarantee that the solution has H -regularity if more assumptions on the medium coeffi-cients, the initial fields and the noise are employed; see Proposition 2.2.In order to inherit the stochastic symplectic structure, we apply the midpoint scheme (3.1) todiscretize (1.1) in time in Section 3. The error is measured in L ( Ω ; V ) , and gives a bound oforder k / D ( M k ) with k ∈ { , } . It is also shown that the divergence conser-vation laws (1.1c) are preserved numerically by the semidiscretization (3.1) in time.We discretize the temporal semidiscretization (3.1) further in space using a dG method, andthen it results the fully discrete method (5.1), called the symplectic dG full discretization; seealso Section 4 for the treatment of the dG approximation of stochastic Maxwell equations. Werefer interested readers to [17] for the application of dG methods to the time-harmonic stochas-tic Maxwell equations with color noise, to [3] for the application to stochastic Helmholtz-typeequation, to [1] for the application to stochastic Allen-Cahn equation, to [2] for the applicationto the semi-linear stochastic wave equation, to [14] for the application to stochastic conservationlaws, and to [4] for the application of a symplectic local dG method to stochastic Schr¨odinger YMPLECTIC DG FULL DISCRETIZATION 3 equation. Since the highest regularity of stochastic Maxwell equations that can be guaranteed isin H , the dG space is taken to be the set of piecewise linear functions. The upwind fluxes areutilized, due to the higher convergence order than the central fluxes; see [11] for the deterministiccase. It is shown in Theorem 4.1 that the mean-square convergence order of the dG approxima-tion (4.4) is of k − / L p ( Ω ; C ([ , T ] ; H k ( D ) )) with k ∈ { , } . This convergence analysis is presented in a form applied also to the full discretiza-tion (5.1), which is stated in Section 5. We also show that the divergence properties (1.1c) arepreserved numerically in a weak sense by the spatial semidiscretization (4.4) and the full dis-cretization (5.1) in Proposition 4.4 and Proposition 5.1, respectively. Moreover, the asymptoticbehaviors of the exact and numerical solutions of stochastic Maxwell equations with small noiseare investigated in Sections 2-5, respectively.To conclude, the main contribution of this paper is to provide a rigorous error analysis of afull discretization for stochastic Maxwell equations. In particular, we prove that:(i) the exact solution and the numerical solution of temporal semidiscrete method belongto L p ( Ω ; C ([ , T ] ; H k ( D ) )) with k ∈ { , } depending only on the assumptions on thethe medium coefficients, the initial fields and the noise, which have not been reportedbefore to the best of our knowledge;(ii) the mean-square error of the full discretization in L ( Ω ; V ) is of order k / k − / ( k ∈ { , } ) , which retains the convergence order of the upwindfluxes space discretization in the deterministic case.2. P ROPERTIES OF STOCHASTIC M AXWELL EQUATIONS
This section presents the notations and basic results for stochastic Maxwell equations, includ-ing the stochastic symplectic structure, the regularity in L p ( Ω ; C ([ , T ] ; H k ( D ) )) with k ∈ { , } ,and the small noise asymptotic behavior. Throughout this paper, we use C to denote a genericconstant, independent of the step sizes τ and h , which may differ from line to line. Let Γ ± j bethe open faces of D given by x j = a ± j , respectively, for j = , , Preliminaries.
We first collect notations used throughout this paper. We use the stan-dard Sobolev spaces W k , p ( D ) : = W k , p ( D , R ) for k ∈ N , p ∈ [ , ∞ ] , where we denote H k ( D ) = W k , ( D ) . For a real number γ ∈ ( , ) and a normed real vector space V , denote C γ ([ , T ] ; V ) : = { f : [ , T ] → V with k f k C γ ([ , T ] ; V ) < ∞ } the space of all γ -H ¨older continuous functions from [ , T ] to V , where k f k C γ ([ , T ] ; V ) : = sup t ∈ [ , T ] k f ( t ) k V + sup t , t ∈ [ , T ] , t = t k f ( t ) − f ( t ) k V | t − t | γ Stochastic Maxwell equations (1.1) are studied in the the real Hilbert space V = L ( D ) × L ( D ) , endowed with the inner product (cid:28)(cid:18) E H (cid:19) , (cid:18) E H (cid:19)(cid:29) V = R D ( ε E · E + µ H · H ) d x for all ( E ⊤ , H ⊤ ) ⊤ , ( E ⊤ , H ⊤ ) ⊤ ∈ V , and the norm (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) EH (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) V = (cid:2) R D (cid:0) ε | E | + µ | H | (cid:1) d x (cid:3) / , ∀ ( E ⊤ , H ⊤ ) ⊤ ∈ V . This space V is equivalent to the usual L ( D ) space under the assumption (1.2) on the coef-ficients ε and µ . CHUCHU CHEN
In addition we use the Hilbert spaces H ( curl , D ) : = { v ∈ L ( D ) : ∇ × v ∈ L ( D ) } , H ( curl , D ) : = { v ∈ H ( curl , D ) : n × v | ∂ D = } , endowed with the norm k u k = k u k L ( D ) + k ∇ × u k L ( D ) , and H ( div , D ) : = { v ∈ L ( D ) : ∇ · v ∈ L ( D ) } , H ( div , D ) : = { v ∈ H ( div , D ) : n · v | ∂ D = } , endowed with the norm k u k = k u k L ( D ) + k ∇ · u k L ( D ) . After these preparations we introduce the Maxwell operator M = (cid:18) ε − ∇ ×− µ − ∇ × (cid:19) , D ( M ) = H ( curl , D ) × H ( curl , D ) (2.1)on V . By defining u ( t ) = ( E ( t ) ⊤ , H ( t ) ⊤ ) ⊤ , the system (1.1) can be rewritten as a stochasticevolution equation ( d u ( t ) = Mu ( t ) d t − d W ( t ) , u ( ) = u , (2.2)where W ( t ) = ( ε − W e ( t ) ⊤ , µ − W m ( t ) ⊤ ) ⊤ is a Q -Wiener process on V with Q = (cid:18) ε − Q e µ − Q m (cid:19) . In fact, for any a = ( a ⊤ , a ⊤ ) ⊤ , b = ( b ⊤ , b ⊤ ) ⊤ ∈ V , we have E [ h W ( t ) , a i V h W ( t ) , b i V ]= E [( h W e ( t ) , a i U + h W m ( t ) , a i U ) ( h W e ( t ) , b i U + h W m ( t ) , b i U )]= h Q e a , b i U + h Q m a , b i U = h Qa , b i V . Note that E k W ( t ) k V = t (cid:0) k ε − Q e k HS ( U , U ) + k µ − Q m k HS ( U , U ) (cid:1) , and Q still is a symmetric, pos-itive definite operator on V with trace Tr ( Q ) = (cid:0) k ε − Q e k HS ( U , U ) + k µ − Q m k HS ( U , U ) (cid:1) . It is notdifficult to show that the energy of the system (1.1) evolutes linearly with a rate Tr ( Q ) , i.e., E k u ( t ) k V = E k u k V + Tr ( Q ) t . Note that (2.2) is an infinite-dimensional Hamiltonian system. If the coefficients ε , µ areconstants, the canonical form of the infinite-dimensional Hamiltonian system of (2.2) readsd u ( t ) = J − δ H δ u d t + J − δ H δ u d ˜ W e + J − δ H δ u d ˜ W m , (2.3)where J = (cid:18) I − I (cid:19) with I being the identity matrix on R × , ˜ W e = ( ⊤ , W ⊤ e ) ⊤ , ˜ W m =( W ⊤ m , ⊤ ) ⊤ , and H = − R D (cid:0) µ − E · ( ∇ × E ) + ε − H · ( ∇ × H ) (cid:1) d x , H = R D ε − H d x , H = − R D µ − E d x . The phase flow of (2.3) preserves the stochastic symplecticity, i.e., for any t ∈ [ , T ] , ω ( t ) = R D d E ∧ d H d x , P -a.s. We refer to [6] for the discussion on the symplecticity YMPLECTIC DG FULL DISCRETIZATION 5 of stochastic Maxwell equations and the numerical preservation of the symplecticity by thesemidiscrete methods in time.The domain D ( M ) includes the electric boundary condition, but neither the magnetic bound-ary condition nor the divergence conditions. In order to regard all conditions, we define V : = { ( E ⊤ , H ⊤ ) ⊤ ∈ V : ∇ · ( ε E ) = ∇ · ( µ H ) = , n · ( µ H ) = ∂ D } , which is a closed subspaceof V with the inner product and norm being defined the same as in V . We mainly work with therestriction M of M on V . It is known that under (1.2), M : D ( M ) = D ( M ) ∩ V → V is skewadjoint, and thus generates a unitary C -group { S ( t ) } t ∈ R on V . Moreover, since M maps D ( M ) into V , we have D ( M k ) = D ( M k ) ∩ V (cf. [10]).2.2. H -regularity. The H -regularity of the solution is deduced by utilizing the fact that v ∈ H ( curl , D ) ∩ H ( div , D ) belongs to H ( D ) if v × n = or v · n = ∂ D . Moreover,the H -norm of v is dominated by k v k H ( D ) ≤ C (cid:16) k v k L ( D ) + k ∇ × v k L ( D ) + k ∇ · v k L ( D ) (cid:17) , where the constant C depends on the space domain D . Since ∇ · ( ε E ) =
0, we get that ∇ · E = ∇ · (cid:0) ε − ε E (cid:1) = ε − ∇ · ( ε E ) + ∇ ( ε − ) · ( ε E ) = − ε − ∇ε · E belongs to L ( D ) if ε ∈ W , ∞ ( D ) with ε ≥ δ > δ >
0, and analogously for H . That means that k ∇ · E k L ( D ) + k ∇ · H k L ( D ) ≤ C (cid:0) δ , k ε k W , ∞ ( D ) , k µ k W , ∞ ( D ) (cid:1) k ( E , H ) k L ( D ) . Hence, D ( M ) = D ( M ) ∩ V ֒ → H ( D ) , if coefficients ε , µ satisfies certain assumptions as above. Moreover, k ( E , H ) k H ( D ) ≤ C k ( E , H ) k D ( M ) , (2.4)with C : = C (cid:0) δ , k ε k W , ∞ ( D ) , k µ k W , ∞ ( D ) (cid:1) . Proposition 2.1.
Let the assumption (1.2) hold, and let Q ∈ HS ( V , D ( M )) and u ∈ L p ( Ω ; D ( M )) for some p ≥ . Then the equation (2.2) has a unique solution u ∈ L p ( Ω ; C ([ , T ] ; D ( M ))) givenby u ( t ) = S ( t ) u − Z t S ( t − s ) d W ( s ) , (2.5) where u also belongs to C ([ , T ] ; L p ( Ω ; V )) . Assume further that ε , µ ∈ W , ∞ ( D ) , then E (cid:20) sup t ∈ [ , T ] k u ( t ) k pH ( D ) (cid:21) ≤ C E (cid:20) sup t ∈ [ , T ] k u ( t ) k p D ( M ) (cid:21) ≤ C ( + E k u k p D ( M ) ) , (2.6) where C depends on T , δ , k ε k W , ∞ ( D ) , k µ k W , ∞ ( D ) and k Q k HS ( V , D ( M )) .Proof. Since M generates a unitary C -group { S ( t ) } t ∈ R on V , the existence and uniqueness ofthe mild solution u ( t ) of (2.5) on V follows. The estimate on stochastic convolution yields h E (cid:0) sup t ∈ [ , T ] k u ( t ) k p D ( M ) (cid:1)i p ≤ h E (cid:0) k u k p D ( M ) (cid:1)i p + (cid:20) E (cid:18) sup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) Z t S ( t − s ) d W ( s ) (cid:13)(cid:13)(cid:13) p D ( M ) (cid:19)(cid:21) p ≤ C (cid:16) + h E (cid:0) k u k p D ( M ) (cid:1)i p (cid:17) , (2.7)where the constant C depends on T and k Q k HS ( V , D ( M )) .Based on [5, Lemma 3.3] and (2.7), for any 0 ≤ s ≤ t ≤ T , we get k u ( t ) − u ( s ) k L p ( Ω ; V ) ≤ k (cid:0) S ( t − s ) − I (cid:1) u ( s ) k L p ( Ω ; V ) + (cid:13)(cid:13)(cid:13)(cid:13) Z ts S ( t − r ) d W ( r ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( Ω ; V ) CHUCHU CHEN ≤ C ( + k u k L p ( Ω ; D ( M )) )( t − s ) + C ( t − s ) , which leads to k u k C ([ , T ] ; L p ( Ω ; V )) = sup t ∈ [ , T ] k u ( t ) k L p ( Ω ; V ) + sup t = s k u ( t ) − u ( s ) k L p ( Ω ; V ) | t − s | ≤ C . Utilizing the embedding (2.4), the assertion (2.6) follows from (2.7). Thus the proof is fin-ished. (cid:3) H -regularity. In our error analysis we need the solution u of (2.2) taking values in H ( D ) ,which relies on additional regularity properties of D ( M ) = D ( M ) ∩ V and some smoothnessof the coefficients ε and µ . Assume that ε , µ ∈ W , ∞ ( D ) ∩ W , ( D ) , with ε , µ ≥ δ for a constant δ > . (2.8)In fact, for any w = ( E , H ) ∈ D ( M ) , we already have w ∈ H ( D ) from (2.4). Further, M w = − ε − ∇ × (cid:0) µ − ∇ × E (cid:1) − µ − ∇ × (cid:0) ε − ∇ × H (cid:1)! ∈ L ( D ) , and the properties of curl operator lead to ∆ E = − ∇ × ( ∇ × E ) + ∇ ( ∇ · E )= − µ∇ × ( µ − ∇ × E ) − µ − ∇µ × (cid:0) ∇ × E (cid:1) − ∇ ( ε − ∇ε · E ) ∈ L ( D ) , if the coefficients ε , µ satisfy (2.8). Then the H -regularity of E follows from the equivalenceof H -norm and the graph norm of Laplacian ∆ on D under certain mixed boundary conditions,i.e., if there is a unique function v ∈ H Γ ( D ) solving Z D v φ d x + Z D ∇ v · ∇φ d x = Z D f φ d x , for f ∈ L ( D ) and ∀ φ ∈ H Γ ( D ) , then the solution v ∈ H ( D ) ∩ H Γ ( D ) satisfies v − ∆ v = f on D , ∂ n v = ∂ D \ Γ , and k v k H ( D ) ≤ C (cid:16) k v k L ( D ) + k ∆ v k L ( D ) (cid:17) with the constant C dependingon D . Here for a union Γ ⊆ ∂ D of some faces of D , H Γ ( D ) : = { v ∈ H ( D ) | tr ( v ) = Γ } . Foreach component E j (resp. H j ) of E (resp. H ), the boundary Γ may be taken as Γ ± k ∪ Γ ± ℓ (resp. Γ ± j ) with j , k , ℓ ∈ { , , } and k = ℓ = j . We refer to [10] for more details. Proposition 2.2.
Let Q ∈ HS ( V , D ( M )) , and u ∈ L p ( Ω ; D ( M )) for some p ≥ . Under theassumption (2.8) , the solution (2.5) has the following property E (cid:20) sup t ∈ [ , T ] k u ( t ) k pH ( D ) (cid:21) ≤ C E (cid:20) sup t ∈ [ , T ] k u ( t ) k p D ( M ) (cid:21) ≤ C ( + E k u k p D ( M ) ) , (2.9) where the constant C depends on T , δ , k ε k W , ∞ ( D ) , k ε k W , ( D ) , k µ k W , ∞ ( D ) , k µ k W , ( D ) and k Q k HS ( V , D ( M )) .Proof. We first prove the D ( M ) -regularity of the solution. From (2.5), we get h E (cid:0) sup t ∈ [ , T ] k u ( t ) k p D ( M ) (cid:1)i p ≤ h E (cid:0) k u k p D ( M ) (cid:1)i p (2.10) YMPLECTIC DG FULL DISCRETIZATION 7 + (cid:20) E (cid:18) sup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) Z t S ( t − s ) d W ( s ) (cid:13)(cid:13)(cid:13) p D ( M ) (cid:19)(cid:21) p ≤ C (cid:18) + h E (cid:0) k u k p D ( M ) (cid:1)i p (cid:19) , where the constant C depends on T and k Q k HS ( V , D ( M )) .The first inequality in (2.9) comes from the embedding D ( M ) ֒ → H ( D ) . Thus the proof isfinished by combining (2.10). (cid:3) Small noise asymptotic behavior.
We scale the noise in the system (2.2) by a small pa-rameter √ λ , λ ∈ R + , i.e., ( d u ( t ) = Mu ( t ) d t − √ λ d W ( t ) , u ( ) = u , (2.11)whose mild solution is given by u u , λ ( t ) = S ( t ) u − √ λ R t S ( t − r ) d W ( r ) . Denote the stochasticconvolution W M ( t ) = R t S ( t − r ) d W ( r ) . Then for arbitrary T > W M ( T ) is Gaussian on V withmean 0 and covariance operator Q T : = Cov (cid:0) W M ( T ) (cid:1) = R T S ( r ) QS ∗ ( r ) d r . Lemma 2.1. [9, Proposition 12.10] Assume that X is a Gaussian random variable with distribu-tion µ = N ( , e Q ) on a Hilbert space H. Then the family of random variables { X λ : = √ λ X } λ > (or measures (cid:8) µ λ = L (cid:0) X λ (cid:1)(cid:9) λ > ) satisfies the large deviation principle with the good rate func-tion I ( x ) = k e Q − x k H , x ∈ e Q ( H ) , + ∞ , otherwise , (2.12) where e Q − is the pseudo inverse of e Q . Based on Lemma 2.1, we get the following asymptotic behavior of the solution for (2.11)with small diffusion coefficient, which states that the laws of solutions satisfy the large deviationprinciple with the good rate function (2.13).
Proposition 2.3.
For arbitrary T > and u ∈ V , the family of distributions (cid:8) L (cid:0) u u , λ ( T ) (cid:1)(cid:9) λ > satisfies the large deviation principle with the good rate functionI u T ( v ) = k Q − T (cid:0) v − S ( T ) u (cid:1) k V , v − S ( T ) u ∈ Q T ( V ) , + ∞ , otherwise , (2.13) where Q − T is the pseudo inverse of Q T .Proof. We define a process Y λ ( t ) = u u , λ ( t ) − S ( t ) u , which satisfies (2.11) with initial data Y λ ( ) =
0. This means that Y λ ( t ) = √ λ W M ( t ) . Then by the large deviation principle for Gauss-ian measures (Lemma 2.1), it follows that the good rate function of { Y λ ( T ) } λ > is given by I T ( v ) = k Q − T v k V , v ∈ Q T ( V ) , + ∞ , otherwise . (2.14) CHUCHU CHEN
In order to give the rate function of { u u , λ ( T ) } λ > based on (2.14), we use the definition of largedeviation principle. Let A ∈ B ( V ) be closed. Then A − { S ( T ) u } still is closed in B ( V ) andhence lim sup λ → h λ ln P { u u , λ ( T ) ∈ A } i = lim sup λ → h λ ln P { Y λ ( T ) ∈ A − { S ( T ) u }} i ≤ − inf v ∈ A −{ S ( T ) u } I T ( v ) = − inf v ∈ A I T ( v − S ( T ) u ) = : − inf v ∈ A I u T ( v ) . In a similar way we can check that for any open B ∈ B ( V ) ,lim inf λ → h λ ln P { u u , λ ( T ) ∈ B } i ≥ − inf v ∈ B I T ( v − S ( T ) u ) = − inf v ∈ B I u T ( v ) . Since I u T fulfills the same properties as I T , i.e. I u T is a good rate function, the proof is thuscompleted. (cid:3) Remark 2.1.
If Q commutes with M, then Q T ( V ) = Q ( V ) . In fact, Q T = R T S ( r ) QS ∗ ( r ) d r = T Q .
3. T
EMPORAL SEMIDISCRETIZATION BY STOCHASTIC SYMPLECTIC METHOD
In this section, we study the semidiscretization in time of (2.2) by a midpoint scheme, whichpreserves the stochastic symplectic structure. The temporal semidiscretizations by a class ofstochastic symplectic Runge-Kutta methods have been studied in [6]. It is shown in there thatthe methods are convergent with order one in mean-square sense, if the solution has regularityin D ( M ) .For the time interval [ , T ] , we introduce the uniform partition 0 = t < t < . . . < t N = T . Let τ = T / N , and ∆ W n + = W ( t n + ) − W ( t n ) , n = , , . . . , N −
1. Applying the midpoint scheme to(2.2) in temporal direction yields u n + = u n + τ ( Mu n + Mu n + ) − ∆ W n + , (3.1)which can also be written as ε E n + = ε E n + τ ( ∇ × H n + ∇ × H n + ) − ∆ W n + e , (3.2a) µ H n + = µ H n − τ ( ∇ × E n + ∇ × E n + ) − ∆ W n + m . (3.2b)This scheme preserves the stochastic symplectic structure numerically, which is stated asfollows. Proposition 3.1. [6, Theorem 4.3] Let ε , µ be constants. Under a zero boundary condition, thetemporal semidiscretization (3.1) preserves the discrete stochastic symplectic structure ω n + = R D d E n + ∧ d H n + d x = R D d E n ∧ d H n d x = ω n , P -a.s. The divergence conservation laws (1.1c) can be preserved numerically by the temporal semidis-cretization (3.1).
Proposition 3.2.
For the temporal semidiscretization (3.1) , if Qh ∈ V for any h ∈ V , then forany n = , , . . . , N − , ∇ · ( ε E n + ) = ∇ · ( ε E n ) , ∇ · ( µ H n + ) = ∇ · ( µ H n ) , P -a.s. YMPLECTIC DG FULL DISCRETIZATION 9
Proof.
The proof follows from the identity ∇ · ( ∇ × U ) = U : R → R . (cid:3) The solution of the temporal semidiscretization (3.1) also has the same regularity as the exactsolution of (2.2), by using the embeddings D ( M ) ֒ → H ( D ) and D ( M ) ֒ → H ( D ) . They arestated below without the proof. Proposition 3.3.
Under the conditions of Proposition 2.1, the solution of the temporal semidis-cretization (3.1) has regularity in H ( D ) , and max ≤ n ≤ N E k u n k pH ( D ) ≤ C ( + E k u k p D ( M ) ) , (3.3) where the constant C depends on T , δ , k ε k W , ∞ ( D ) , k µ k W , ∞ ( D ) and k Q k HS ( V , D ( M )) . Proposition 3.4.
Under the conditions of Proposition 2.2, the solution of the temporal semidis-cretization (3.1) has regularity in H ( D ) , and max ≤ n ≤ N E k u n k pH ( D ) ≤ C ( + E k u k p D ( M ) ) , (3.4) where the constant C depends on T , δ , k ε k W , ∞ ( D ) , k ε k W , ( D ) , k µ k W , ∞ ( D ) , k µ k W , ( D ) and k Q k HS ( V , D ( M )) . Let S τ = (cid:0) I − τ M (cid:1) − (cid:0) I + τ M (cid:1) and T τ = (cid:0) I − τ M (cid:1) − . The mild version of (3.1) reads u n + = S τ u n − T τ ∆ W n + = S n + τ u − n + ∑ j = S n + − j τ T τ ∆ W j . (3.5) Lemma 3.1.
There exists a positive constant C independent of τ such that k I − T τ k L ( D ( M ) , V ) ≤ C τ . Proof.
We define e v = T τ v for any v ∈ D ( M ) , which means that e v = v + τ M e v . Taking innerproduct with e v yields h k e v k V − k v k V + k e v − v k V i = τ h M e v , e v i V = . Hence k e v k V = k T τ v k V ≤k v k V leads to k T τ k L ( V , V ) ≤ k e v − v k V ≤ C τ k v k D ( M ) . In fact, k e v − v k V = τ k M e v k V = τ k T τ Mv k V ≤ τ k v k D ( M ) . Therefore the proof is finished. (cid:3)
For the semigroups S ( t n ) and S n τ , we have the following estimates. Lemma 3.2.
For any integer n ∈ { , . . . , N } , there exists a positive constant C independent of τ such that k S ( t n ) − S n τ k L ( D ( M k ) , V ) ≤ C τ k / with k ∈ { , } . Proof.
In order to estimate the error of semigroups, we denote v ( t ) = S ( t ) v and v k = S k τ v .Then { v ( t ) } t ∈ [ , T ] is the exact solution of dd t v = Mv , v ( ) = v , while { v k } ≤ k ≤ N is the solutionof v k = v k − + τ (cid:0) Mv k − + Mv k (cid:1) , v = v . Note that v ( t k ) = v ( t k − ) + R t k t k − Mv ( s ) d s leads to e k = e k − + τ (cid:0) Me k − + Me k (cid:1) + Z t k t k − h Mv ( s ) − Mv ( t k − ) − Mv ( t k ) i d s , where e k = v ( t k ) − v k . Applying h· , e k + e k − i V to both sides of the above equation, and usingthe skew-adjoint property of the operator M , we get k e k k V = k e k − k V + Z t k t k − D Mv ( s ) − Mv ( t k − ) − Mv ( t k ) , e k + e k − E V d s = k e k − k V − Z t k t k − D Z st k − Mv ( r ) d r − Z t k s Mv ( r ) d r , Me k + Me k − E V d s (3.6) ≤ k e k − k V + C τ (cid:16) sup t ∈ [ , T ] k v ( t ) k D ( M ) + max ≤ k ≤ N k v k k D ( M ) (cid:17) ≤ k e k − k V + C τ k v k D ( M ) , which yields max ≤ k ≤ N k e k k V = max ≤ k ≤ N k (cid:0) S ( t k ) − S k τ (cid:1) v k V ≤ C τ k v k D ( M ) .In the other hand, based on (3.6), k e k k V = k e k − k V − Z t k t k − D Z st k − Mv ( r ) d r − Z t k s Mv ( r ) d r , Me k + Me k − E V d s = k e k − k V + Z t k t k − D(cid:16) Z st k − Z rt k − − Z t k s Z rt k − (cid:17) Mv ( ξ ) d ξ d r , M ( e k + e k − ) E V d s ≤ k e k − k V + C τ k v k D ( M ) , which yields max ≤ k ≤ N k e k k V = max ≤ k ≤ N k (cid:0) S ( t k ) − S k τ (cid:1) v k V ≤ C τ k v k D ( M ) . Therefore, the proof isfinished. (cid:3) Theorem 3.1.
Let Q ∈ HS ( V , D ( M k )) and u ∈ L ( Ω ; D ( M k )) with k ∈ { , } . For the tem-poral semidiscretization (3.1) , we have max ≤ n ≤ N (cid:0) E k u ( t n ) − u n k V (cid:1) / ≤ C τ k / , for k ∈ { , } , (3.7) where the positive constant C depends on T , k u k L ( Ω ; D ( M k )) and k Q k HS ( V , D ( M k )) , but indepen-dent of τ and n.Proof. From the mild solutions (2.5) and (3.5), we use Itˆo isometry to get E k u ( t n ) − u n k V ≤ E k (cid:0) S ( t n ) − S n τ (cid:1) u k V + E (cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = Z t j t j − (cid:0) S ( t n − r ) − S n − j τ T τ (cid:1) d W (cid:13)(cid:13)(cid:13)(cid:13) V = E k (cid:0) S ( t n ) − S n τ (cid:1) u k V + n ∑ j = Z t j t j − (cid:13)(cid:13)(cid:13)(cid:0) S ( t n − r ) − S n − j τ T τ (cid:1) Q (cid:13)(cid:13)(cid:13) HS ( V , V ) d r . The first term on the right-hand side is estimated by Lemma 3.2, and the second term on theright-hand side can be estimated by, for r ∈ [ t j − , t j ] , (cid:13)(cid:13)(cid:13)(cid:0) S ( t n − r ) − S n − j τ T τ (cid:1) Q (cid:13)(cid:13)(cid:13) HS ( V , V ) ≤ k S ( t n − t j ) (cid:0) S ( t j − r ) − I (cid:1) Q k HS ( V , V ) + k (cid:0) S ( t n − t j ) − S n − j τ (cid:1) Q k HS ( V , V ) + k S n − j τ (cid:0) I − T τ (cid:1) Q k HS ( V , V ) ≤ C τ k Q k HS ( V , D ( M )) + C τ k / k Q k HS ( V , D ( M k )) , for k ∈ { , } , where in the last step, we use Lemmas 3.1-3.2 and [5, Lemma 3.3]. Combining them together,we finish the proof. (cid:3) Applying the midpoint scheme to discretize the system (2.11) with small noise, we get that u N = S N τ u − √ λ ∑ Nj = S N − j τ T τ ∆ W j . Let W M ; N : = ∑ Nj = S N − j τ T τ ∆ W j . Then it is Gaussian on V YMPLECTIC DG FULL DISCRETIZATION 11 with mean 0 and covariance operator Q T ; N : = Cov ( W M ; N ) = τ∑ Nj = (cid:0) S N − j τ T τ (cid:1) Q ( S N − j τ T τ (cid:1) ∗ . Anal-ogously, as in Proposition 2.3, we get the following result.
Proposition 3.5.
For integer N > and u ∈ V , the family of distributions { L (cid:0) u N ; u , λ (cid:1) } λ > satisfies the large deviation principle with the good rate functionI u T , N ( v ) = ( k (cid:0) Q T ; N (cid:1) − (cid:0) v − S N τ u (cid:1) k V , v − S N τ u ∈ (cid:0) Q T ; N (cid:1) ( V ) , + ∞ , otherwise . (3.8) Remark 3.1.
If Q commutes with M, then (cid:0) Q T ; N (cid:1) ( V ) = (cid:0) T τ Q (cid:1) ( V ) ⊂ Q ( V ) . In fact, Q T ; N = τ∑ Nj = (cid:0) S N − j τ T τ (cid:1) Q ( S N − j τ T τ (cid:1) ∗ = τ NT τ QT ∗ τ = T T τ QT ∗ τ yields the assertion. Proposition 3.6.
Assume that Q commutes with M, and v , u ∈ (cid:0) T τ Q (cid:1) ( V ) , then there is aconstant C depending on T , k Q − v k D ( M ) and k Q − u k D ( M ) such that (cid:12)(cid:12)(cid:12) I u T ( v ) − I u T , N ( v ) (cid:12)(cid:12)(cid:12) ≤ C τ . In addition, if Q − v, Q − u ∈ D ( M ) , then there is a constant C depending on T , k Q − v k D ( M ) and k Q − u k D ( M ) such that (cid:12)(cid:12)(cid:12) I u T ( v ) − I u T , N ( v ) (cid:12)(cid:12)(cid:12) ≤ C τ . Proof.
Note that under the conditions of this proposition, I u T ( v ) = T (cid:13)(cid:13)(cid:13) Q − ( v − S ( T ) u ) (cid:13)(cid:13)(cid:13) V , I u T , N ( v ) = T (cid:13)(cid:13)(cid:13) Q − T − τ (cid:0) v − S N τ u (cid:1)(cid:13)(cid:13)(cid:13) V . Thus, (cid:12)(cid:12)(cid:12) I u T ( v ) − I u T , N ( v ) (cid:12)(cid:12)(cid:12) (3.9) = T (cid:12)(cid:12)(cid:12)D Q − ( v − S ( T ) u ) + Q − T − τ (cid:0) v − S N τ u (cid:1) , Q − ( v − S ( T ) u ) − Q − T − τ (cid:0) v − S N τ u (cid:1) E V (cid:12)(cid:12)(cid:12) ≤ C (cid:13)(cid:13)(cid:13) Q − ( v − S ( T ) u ) − Q − T − τ (cid:0) v − S N τ u (cid:1)(cid:13)(cid:13)(cid:13) V ≤ C h(cid:13)(cid:13)(cid:13) Q − ( I − T − τ ) ( v − S ( T ) u ) (cid:13)(cid:13)(cid:13) V + (cid:13)(cid:13)(cid:13) Q − T − τ (cid:0) S ( T ) − S N τ (cid:1) u (cid:13)(cid:13)(cid:13) V i , where the constant C depends on T , k Q − T − τ v k V , k Q − T − τ u k V . Since I − T − τ = τ M , (cid:13)(cid:13)(cid:13) Q − ( I − T − τ ) ( v − S ( T ) u ) (cid:13)(cid:13)(cid:13) V ≤ C ( k Q − Mv k V , k Q − Mu k V ) τ . And for the second term on the right-hand side of (3.9), (cid:13)(cid:13)(cid:13) Q − T − τ (cid:0) S ( T ) − S N τ (cid:1) u (cid:13)(cid:13)(cid:13) V ≤ (cid:13)(cid:13)(cid:13) Q − (cid:0) S ( T ) − S N τ (cid:1) u (cid:13)(cid:13)(cid:13) V + τ (cid:13)(cid:13)(cid:13) Q − M (cid:0) S ( T ) − S N τ (cid:1) u (cid:13)(cid:13)(cid:13) V ≤ (cid:13)(cid:13)(cid:13) Q − (cid:0) S ( T ) − S N τ (cid:1) u (cid:13)(cid:13)(cid:13) V + C ( k Q − Mu k V ) τ . Lemma 3.2 yields the conclusion and thus the proof is finished. (cid:3)
4. S
PATIAL SEMIDISCRETIZATION BY D G METHOD
In this section, we investigate the semidiscretization of the stochastic Maxwell equations(2.2) in space by the dG method with the upwind fluxes, including the properties of the discreteMaxwell operator, the well-posedness of the spatial semidiscretization, the preservation of thedivergence properties in a weak sense, and the mean-square error estimate of the semidiscretemethod in space.4.1.
Discrete Maxwell operator.
The notations and properties of the discrete Maxwell op-erator are based on [11]. Let T h = { K } be a simplicial, shape- and contact-regular mesh ofthe domain D consisting of elements K , i.e., D = S K . The index h refers to the maximumdiameter of all elements of T h . The dG space with respect to the mesh T h is taken to bethe set of piecewise linear functions, i.e., V h : = P ( T h ) : = { v h ∈ L ( D ) : v h | K ∈ P ( K ) } , where P ( K ) denotes the set of continuous piecewise polynomials of degree ≤
1. In general, V h D ( M ) . The set of faces is denoted by G h = G int h ∪ G ext h , where G int h and G ext h consist of allinterior and all exterior faces, respectively. By n F we denote the unit normal of a face F ∈ G int h ,where the orientation of n F is fixed once and forever for each inner face. And for a bound-ary face F ∈ G ext h , n F is an outward normal vector. The broken Sobolev spaces are defined by H k ( T h ) : = { v ∈ L ( D ) : v | K ∈ H k ( K ) for all K ∈ T h } , k ∈ N , with seminorm and norm being | v | H k ( T h ) : = ∑ K ∈ T h | v | H k ( K ) and k v k H k ( T h ) : = k ∑ j = | v | H j ( T h ) , respectively. Note that H k ( D ) ⊂ H k ( T h ) . Assumption 4.1.
Assume that π h : V → V h is the orthogonal projection, defined by, for everyv ∈ V , h v − π h v , u h i V = for all u h ∈ V h . (4.1) Moreover, for all v ∈ H s ( T h ) with integer s ≤ , it holds that k v − π h v k V ≤ Ch s | v | H s ( T h ) , (4.2) and ∑ F ∈ G h k v − π h v k L ( F ) ≤ Ch s − | v | H s ( T h ) , (4.3) where the constant C is independent of h. Remark 4.1. (i)
For the projection operator π h in Assumption 4.1, it is not difficult to getthat k π h v k V ≤ k v k V . (ii) Suppose that µ K : = µ | K and ε K : = ε | K are constants for each K ∈ T h , then the usualL -orthogonal projection π h on P ( T h ) satisfies Assumption 4.1, where the projectionacts componentwise for vector fields. Define by [[ v ]] F : = (cid:0) v K F (cid:1) | F − (cid:0) v K (cid:1) | F the jump of v on an interior face F with normal vector n F pointing from K to K F . The Maxwell operator discretized by a dG method with the upwindfluxes is defined as follows. Definition 4.1.
Given u h = ( E ⊤ h , H ⊤ h ) ⊤ , v h = ( ψ ⊤ h , φ ⊤ h ) ⊤ ∈ V h , the discrete Maxwell operatorM h : V h → V h is given as h M h u h , v h i V : = ∑ K (cid:16) h ∇ × H h , ψ h i L ( K ) − h ∇ × E h , φ h i L ( K ) (cid:17) YMPLECTIC DG FULL DISCRETIZATION 13 + ∑ F ∈ G int h (cid:16) h n F × [[ H h ]] F , β K ψ K + β K F ψ K F i L ( F ) − h n F × [[ E h ]] F , α K φ K + α K F φ K F i L ( F ) − γ F h n F × [[ E h ]] F , n F × [[ ψ h ]] F i L ( F ) − δ F h n F × [[ H h ]] F , n F × [[ φ h ]] F i L ( F ) (cid:17) + ∑ F ∈ G ext h (cid:16) h n × E h , φ h i L ( F ) − γ F h n × E h , n × ψ h i L ( F ) (cid:17) , where α K = C K F ε K F C K F ε K F + C K ε K , β K = C K F µ K F C K F µ K F + C K µ K , γ F = C K F µ K F + C K µ K , δ F = C K F ε K F + C K ε K , with C K = ( ε K µ K ) − / . The discrete Maxwell operator M h is also well-defined as an operator from V h + (cid:0) D ( M ) ∩ H ( T h ) (cid:1) to V h , and has the following properties. Here V h + (cid:0) D ( M ) ∩ H ( T h ) (cid:1) : = { v h + u : v h ∈ V h , u ∈ D ( M ) ∩ H ( T h ) } . We refer to [11, Lemmas 4.3-4.5] for proofs. Proposition 4.1. (i)
For u ∈ D ( M ) ∩ H ( T h ) , we have M h u = π h Mu. (ii)
For all u h = ( E ⊤ h , H ⊤ h ) ⊤ ∈ V h , we have h M h u h , u h i V = − ∑ F ∈ G inth (cid:16) γ F k n F × [[ E h ]] F k L ( F ) + δ F k n F × [[ H h ]] F k L ( F ) (cid:17) − ∑ F ∈ G exth γ F k n F × E h k L ( F ) ≥ . In particular, M h is dissipative on V h . (iii) For u = ( E ⊤ , H ⊤ ) ⊤ ∈ V h + (cid:0) D ( M ) ∩ H ( T h ) (cid:1) and v h = ( ψ ⊤ h , φ ⊤ h ) ⊤ ∈ V h , we have h M h u , v h i V = ∑ K (cid:16) h H , ∇ × ψ h i L ( K ) − h E , ∇ × φ h i L ( K ) (cid:17) + ∑ F ∈ G inth (cid:16) h β K H K F + β K F H K − γ F n F × [[ E ]] F , n F × [[ ψ h ]] F i L ( F ) − h α K E K F + α K F E K + δ F n F × [[ H ]] F , n F × [[ φ h ]] F i L ( F ) (cid:17) + ∑ F ∈ G exth h H , n F × ψ h i L ( F ) − γ F h n F × E , n F × ψ h i L ( F ) . Semidiscrete method in space.
After discretizing (2.2) by a dG method with the upwindfluxes, we end up with the spatial semidiscretization ( d u h ( t ) = M h u h ( t ) d t − π h d W ( t ) , u h ( ) = π h u , (4.4)where M h is the discrete Maxwell operator in Definition 4.1, and u h ( t ) ∈ V h is an approximationof the exact solution u ( t ) ∈ V . Notice that the equation (4.4) actually is a finite dimensional stochastic differential equation.In fact, let { φ , . . . , φ N h } be a basis for V h . Utilizing this basis, the semidiscrete problem (4.4) inspace can be rewritten as, for j = , . . . , N h , ( d h u h ( t ) , φ j i V = h M h u h ( t ) , φ j i V d t − h φ j , d W ( t ) i V , h u h ( ) , φ j i V = h u , φ j i V . (4.5)Since u h ( t ) ∈ L ( Ω ; V h ) , we get u h ( t ) = N h ∑ ℓ = u [ ℓ ] ( t ) φ ℓ . Denoting A = (cid:16) h φ ℓ , φ j i V (cid:17) j ,ℓ ∈ R N h × N h , B = (cid:16) h M h φ ℓ , φ j i V (cid:17) j ,ℓ ∈ R N h × N h , u ( t ) = ( u [ ] ( t ) , . . . , u [ N h ] ( t )) ⊤ ∈ R N h , u = ( h u , φ i V , . . . , h u , φ N h i V ) ⊤ ∈ R N h and W ( t ) = ( W [ ] ( t ) , . . . , W [ N h ] ( t )) ⊤ ∈ R N h with W [ j ] ( t ) = h φ j , W ( t ) i V , we obtain the systemof stochastic ordinary differential equations on R N h for (4.4), ( A d u ( t ) = B u ( t ) d t − d W ( t ) , A u ( ) = u . (4.6)Notice that the components of W ( t ) are correlated with E (cid:0) W [ j ] ( t ) W [ ℓ ] ( t ) (cid:1) = E ( h φ j , W ( t ) i V h φ ℓ , W ( t ) i V ) = t h Q φ j , φ ℓ i V , ∀ j , ℓ = , . . . , N h . Proposition 4.2.
The spatially semidiscrete problem (4.4) is well-posed, i.e., there is a uniquesolution u h ∈ L ( Ω ; C ([ , T ] ; V h )) given byu h ( t ) = e tM h u h ( ) − Z t e ( t − s ) M h π h d W ( s ) . (4.7) Moreover, we have E h sup t ∈ [ , T ] k u h ( t ) k V i ≤ C ( + E k u k V ) , (4.8) where the constant C depends on T and Tr ( Q ) .Proof. Note that I − M h : V h → V h is injective and surjective, and thus Ran ( I − M h ) = V h . Sincethe discrete operator M h is dissipative on V h , it generates a contraction semigroup. Therefore,the unique solution of (4.4) is given by (4.7).The estimate in (4.8) is obtained by the triangle inequality and the estimate on stochasticconvolution E h sup t ∈ [ , T ] k u h ( t ) k V i ≤ E h sup t ∈ [ , T ] k e tM h u h ( ) k V i + E h sup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) Z t e ( t − s ) M h π h d W ( s ) (cid:13)(cid:13)(cid:13) V i ≤ E k u h ( ) k V + T E k π h Q k HS ( V , V ) ≤ C ( + E k u k V ) , where in the last step we use the property k π h u k V ≤ k u k V of the projection operator. Thus theproof is finished. (cid:3) It is not difficult to observe that W M ; h ( t ) = R t e ( t − s ) M h π h d W ( s ) is Gaussian on V h with mean0 and covariance operator Q T , h : = Cov (cid:0) W M ; h ( T ) (cid:1) = Z T (cid:0) e rM h π h (cid:1) Q (cid:0) e rM h π h (cid:1) ∗ d r . YMPLECTIC DG FULL DISCRETIZATION 15
Applying the dG method to discretize the spatial direction of the small noise system (2.11), wedenote by { L (cid:0) u u , λ h ( T ) (cid:1) } λ > the laws of the semidiscrete solutions. The asymptotic behaviorof { L (cid:0) u u , λ h ( T ) (cid:1) } λ > is similar to that of { L (cid:0) u u , λ ( T ) (cid:1) } λ > in Proposition 2.3, which is statedbelow. Proposition 4.3.
For arbitrary T > and u ∈ V , the family of distributions n L (cid:0) u u , λ h ( T ) (cid:1)o λ > satisfies the large deviation principle with the good rate functionI u T , h ( v ) = k Q − T , h (cid:0) v − e TM h π h u (cid:1) k V , v − e TM h π h u ∈ Q T , h ( V h ) , + ∞ , otherwise , (4.9) where Q − T , h is the pseudo inverse of Q T , h . Discrete divergence conservation property. If u ∈ V and Q ∈ HS ( V , V ) , the exactsolution u ( t ) of the stochastic Maxwell equations (2.2) possesses the divergence relations (1.1c): ∇ · ( ε E ( t )) = ∇ · ( ε H ( t )) =
0. However, for the spatial semidiscretization (4.4), we provethat the divergence relations is preserved numerically in the following discrete weak sense.Define the test space X h ⊂ H ( D ) as X h : = { v ∈ C ( ¯ D ) : v h | K ∈ P ( K ) , K ∈ T h } ∩ H ( D ) . By h· , ·i − we denote the duality product between H − ( D ) and H ( D ) , in which h ∇ · E , ψ i − = −h E , ∇ψ i L ( D ) , ∀ E ∈ L ( D ) , ψ ∈ H ( D ) . Proposition 4.4.
Let u ∈ V and Q ∈ HS ( V , V ) . The solution ( E h ( t ) , H h ( t )) of the spatiallysemidiscrete problem (4.4) satisfies: ∀ t ∈ [ , T ] , and ∀ φ ∈ X h , h ∇ · ( ε E h ( t )) , φ i − = h ∇ · ( µ H h ( t )) , φ i − = , P -a.s.Proof. For ψ , φ ∈ X h , using the definition of the duality product h· , ·i − , we get (cid:28)(cid:18) ∇ · ( ε E h ( t )) ∇ · ( µ H h ( t )) (cid:19) , (cid:18) ψφ (cid:19)(cid:29) − = h ∇ · ( ε E h ( t )) , ψ i − + h ∇ · ( ε H h ( t )) , φ i − = −h ε E h ( t ) , ∇ψ i L ( D ) − h ε H h ( t ) , ∇φ i L ( D ) = − (cid:28)(cid:18) E h ( t ) H h ( t ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V . Using (4.4) we obtain (cid:28)(cid:18) E h ( t ) H h ( t ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28)(cid:18) E h ( ) H h ( ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V + Z t (cid:28) M h (cid:18) E h ( s ) H h ( s ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V d s − (cid:28) π h (cid:18) ε − W e ( t ) µ − W m ( t ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V . For the first and third terms on the right-hand side, we utilize the property (4.1) of projectionand the fact that (cid:18) ∇ψ∇φ (cid:19) ∈ V h to get (cid:28)(cid:18) E h ( ) H h ( ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28) π h (cid:18) E ( ) H ( ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28)(cid:18) E ( ) H ( ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = , and (cid:28) π h (cid:18) ε − W e ( t ) µ − W m ( t ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28)(cid:18) ε − W e ( t ) µ − W m ( t ) (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = h W e ( t ) , ∇ψ i L ( D ) + h W m ( t ) , ∇φ i L ( D ) = −h ∇ · W e ( t ) , ψ i − − h ∇ · W m ( t ) , φ i − = . Using Proposition 4.1 (iii), the second term on the right-hand side equals to zero, since for anyfunction ϕ ∈ X h , we have ∇ × ∇ϕ = , n F × [[ ∇ϕ ]] F = for F ∈ G int h and n × ∇ϕ = on ∂ D .Therefore, the conclusion of this proposition comes from taking φ = ψ =
0, respectively. (cid:3)
Remark 4.2.
The projection of the exact solution of (2.2) has the same property, ∀ t ∈ [ , T ] ,and ∀ φ ∈ X h , h ∇ · π h ( ε E ( t )) , φ i − = h ∇ · π h ( µ H ( t )) , φ i − = , P -a.s.In fact, since ∇φ ∈ V h , we have h ∇ · π h ( ε E ( t )) , φ i − = h π h ( ε E ( t )) , ∇φ i L ( D ) = h ε E ( t ) , ∇φ i L ( D ) = h ∇ · ( ε E ( t )) , φ i − = . Error estimate of spatial semidiscretization.
To investigate the error of the spatial semidis-cretization (4.4), we apply the projection π h to the continuous problem (2.2) and use Proposition4.1 (i) to get d π h u ( t ) = M h u ( t ) d t − π h d W ( t ) , π h u ( ) = π h u . (4.10)We define the error e ( t ) = u h ( t ) − u ( t ) = (cid:0) u h ( t ) − π h u ( t ) (cid:1) − (cid:0) u ( t ) − π h u ( t ) (cid:1) = : e h ( t ) − e π ( t ) .The mean-square error estimate of the spatial semidiscretization (4.4) is given in the followingtheorem. Theorem 4.1.
Let u ∈ C ([ , T ] ; L ( Ω ; H k ( D ) )) with k ∈ { , } be the solution of (2.2) and letu h ∈ C ([ , T ] ; L ( Ω ; V h )) be the solution of (4.4) . Then there is a constant C independent of hsuch that sup t ∈ [ , T ] (cid:16) E k u h ( t ) − u ( t ) k V (cid:17) ≤ Ch k − for k ∈ { , } . Proof.
For the part e π ( t ) , by using (4.2), we have E k e π ( t ) k V = E k u ( t ) − π h u ( t ) k V ≤ Ch k E | u ( t ) | H k ( D ) . (4.11)For the part e h ( t ) , we subtract (4.10) from (4.4) to get d e h ( t ) = M h e h ( t ) d t − M h e π ( t ) d t , e h ( ) = . Then we obtain, for any t ∈ [ , T ] ,12 k e h ( t ) k V − Z t h M h e h ( s ) , e h ( s ) i V d s = − Z t h M h e π ( s ) , e h ( s ) i V d s . For the term on the right-hand side, noticing e h ( s ) ∈ V h , e π ( s ) ∈ V h + ( D ( M ) ∩ H ( D ) ) , weuse Proposition 4.1 (iii) to obtain |h M h e π , e h i V | = ∑ K (cid:16) h e H π , ∇ × e E h i L ( K ) − h e E π , ∇ × e H h i L ( K ) (cid:17) + ∑ F ∈ G inth (cid:16) h β K e H π , K F + β K F e H π , K − γ F n F × [[ e E π ]] F , n F × [[ e E h ]] F i L ( F ) − h α K e E π , K F + α K F e E π , K + δ F n F × [[ e H π ]] F , n F × [[ e H h ]] F i L ( F ) (cid:17) YMPLECTIC DG FULL DISCRETIZATION 17 + ∑ F ∈ G exth h e H π , n F × e E h i L ( F ) − γ F h n F × e E π , n F × e E h i L ( F ) , where e π = (cid:0) ( e E π ) ⊤ , ( e H π ) ⊤ (cid:1) ⊤ and e h = (cid:0) ( e E h ) ⊤ , ( e H h ) ⊤ (cid:1) ⊤ . The property of the projection π h leads to h e H π , ∇ × e E h i L ( K ) = h e E π , ∇ × e H h i L ( K ) =
0. Then using Cauchy-Schwarz and Young’sinequalities, we have h M h e π ( s ) , e h ( s ) i V ≤ ∑ F ∈ G exth γ F k n F × e E h k L ( F ) + ∑ F ∈ G inth (cid:16) γ F k n F × [[ e E h ]] F k L ( F ) + δ F k n F × [[ e H h ]] F k L ( F ) (cid:17) + ∑ F ∈ G inth (cid:16) γ F k β K e H π , K F + β K F e H π , K − γ F n F × [[ e E π ]] F k L ( F ) + δ F k α K e E π , K F + α K F e E π , K + δ F n F × [[ e H π ]] F k L ( F ) (cid:17) (4.12) + ∑ F ∈ G exth (cid:16) γ F k e H π k L ( F ) + γ F k n F × e E π k L ( F ) (cid:17) ≤ − h M h e h ( s ) , e h ( s ) i V + Ch k − | u ( s ) | H k ( D ) , where in the last step, we use the equality in (ii) of Proposition 4.1 and the inequality (4.3).Hence, we have12 k e h ( t ) k V − Z t h M h e h ( s ) , e h ( s ) i V d s ≤ Ch k − Z t | u ( s ) | H k ( D ) d s . Proposition 4.1 (ii) yields that the second term on the left-hand side is nonnegative. Then takingexpectation and using Lemmas 2.1 and 2.2, we get sup t ∈ [ , T ] E k e h ( t ) k V ≤ Ch k − R T E | u ( s ) | H k ( D ) d s , which combines with (4.11) completes the proof. (cid:3)
5. F
ULL DISCRETIZATION OF STOCHASTIC M AXWELL EQUATIONS
In this section, we consider the full discretization of stochastic Maxwell equations (2.2) byapplying the midpoint scheme in time and the dG method with the upwind fluxes in space: u n + h = u nh + τ (cid:0) M h u nh + M h u n + h (cid:1) − π h ∆ W n + , (5.1)with u h = π h u . Utilizing the basis of V h in Section 4, the fully discrete method (5.1) can berewritten as the midpoint scheme for (4.6), A u n + = A u n + τ (cid:0) B u n + B u n + (cid:1) − ∆ W n + . Following the proof of Proposition 4.4, the divergence conservation property (1.1c) is pre-served numerically by the solution of (5.1) in a weak sense.
Proposition 5.1.
Let u ∈ V and Q ∈ HS ( V , V ) . The solution (cid:8) u nh (cid:9) ≤ n ≤ N of the fully discretemethod (5.1) satisfies: ∀ n ∈ { , , . . . , N } , and ∀ φ ∈ X h , h ∇ · ( ε E nh ) , φ i − = h ∇ · ( µ H nh ) , φ i − = , P -a.s. Proof.
For ψ , φ ∈ X h , using the definition of the inner product h· , ·i − , we get (cid:28)(cid:18) ∇ · ( ε E n + h ) ∇ · ( µ H n + h ) (cid:19) , (cid:18) ψφ (cid:19)(cid:29) − = h ∇ · ( ε E n + h ) , ψ i − + h ∇ · ( ε H n + h ) , φ i − = − (cid:28)(cid:18) E n + h H n + h (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V . Using (5.1) we obtain (cid:28)(cid:18) E n + h H n + h (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28)(cid:18) E nh H nh (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V + τ (cid:28) M h (cid:18) E nh + E n + h H nh + H n + h (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V − (cid:28) π h (cid:18) ε − ∆ W n + e µ − ∆ W n + m (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V . Using Proposition 4.1 (iii), the second term on the right-hand side equals to zero, since for anyfunction ϕ ∈ X h , we have ∇ × ∇ϕ = , n F × [[ ∇ϕ ]] F = for F ∈ G int h and n × ∇ϕ = on ∂ D .For the third term on the right-hand side, the property of the projection (4.1), and the fact that (cid:18) ∇ψ∇φ (cid:19) ∈ V h yield (cid:28) π h (cid:18) ε − ∆ W n + e µ − ∆ W n + m (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28)(cid:18) ε − ∆ W n + e µ − ∆ W n + m (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = −h ∇ · ( ∆ W n + e ) , ψ i − − h ∇ · ( ∆ W n + m ) , φ i − = . Thus, (cid:28)(cid:18) E n + h H n + h (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28)(cid:18) E nh H nh (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = · · · = (cid:28)(cid:18) E h H h (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = , where in the last step, we use (cid:28)(cid:18) E h H h (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28) π h (cid:18) E H (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = (cid:28)(cid:18) E H (cid:19) , (cid:18) ∇ψ∇φ (cid:19)(cid:29) V = . Therefore the conclusion of this proposition comes from taking φ = ψ =
0, respectively. (cid:3)
The mild version of the full discretization (5.1) can be rewritten as u n + n = S h , τ u nh − T h , τ π h ∆ W n + , (5.2)where T h , τ = (cid:0) I − τ M h (cid:1) − and S h , τ = (cid:0) I − τ M h (cid:1) − (cid:0) I + τ M h (cid:1) . Lemma 5.1.
For operators T h , τ and S h , τ on V h , the following estimates hold: (i) k T h , τ k L ( V h , V h ) ≤ . (ii) k S nh , τ k L ( V h , V h ) ≤ for any ≤ n ≤ N.Proof.
To prove the assertion (i), we define e v = T h , τ v for any v ∈ V h , which means that e v = v + τ M h e v . Taking the inner product with e v yields h k e v k V − k v k V + k e v − v k V i = τ h M h e v , e v i V ≤ . Hence k e v k V = k T h , τ v k V ≤ k v k V leads to the assertion (i).Similarly, to prove the assertion (ii), we define v nh = S nh , τ v for any v ∈ V h , which means that v ℓ h = v ℓ − h + τ (cid:0) M h v ℓ − h + M h v ℓ h (cid:1) , ℓ = , , . . . , n , with v h = v . Taking the inner product with YMPLECTIC DG FULL DISCRETIZATION 19 ( v ℓ − h + v ℓ h ) yields k v ℓ h k V − k v ℓ − h k V ≤ , and thus k v ℓ h k V ≤ k v ℓ − h k V ≤ . . . ≤ k v h k V = k v k V . Thisleads to the assertion (ii). (cid:3) Proposition 5.2.
There exists a constant C independent of h and τ such that max ≤ n ≤ N E k u nh k V ≤ C ( + E k u k V ) . Proof.
From (5.2), we know that u nh = S nh , τ π h u − ∑ nj = S n − jh , τ T h , τ π h ∆ W j . Taking k · k V -norm onboth sides of the above equation and using the triangle inequality, we get E k u nh k V ≤ E k S nh , τ π h u k V + E (cid:13)(cid:13)(cid:13) n ∑ j = S n − jh , τ T h , τ π h ∆ W j (cid:13)(cid:13)(cid:13) V ≤ E k π h u k V + n ∑ j = E (cid:13)(cid:13) π h ∆ W j (cid:13)(cid:13) V ≤ E k u k V + T Tr ( Q ) , which completes the proof. (cid:3) Let W M ; N , h : = N ∑ j = S N − jh , τ T h , τ π h ∆ W j . Then it is Gaussian on V with mean 0 and covarianceoperator Q T ; N , h : = Cov ( W M ; N , h ) = τ N ∑ j = (cid:0) S N − jh , τ T h , τ π h (cid:1) Q ( S N − jh , τ T h , τ π h (cid:1) ∗ . Applying the fully discrete method to the small noise system (2.11), we denote by { L (cid:0) u N ; u , λ h (cid:1) } λ > the laws of the fully discretizations. The asymptotic behavior of { L (cid:0) u N ; u , λ h (cid:1) } λ > is similar tothat of { L (cid:0) u N ; u , λ (cid:1) } λ > in Proposition 3.5, which is stated below. Proposition 5.3.
For integer N > and u ∈ V , the family of distributions { L (cid:0) u N ; u , λ h (cid:1) } λ > satisfies the large deviation principle with the good rate functionI u T ; N , h ( v ) = ( k (cid:0) Q T ; N , h (cid:1) − (cid:0) v − S Nh , τ π h u (cid:1) k V , v − S Nh , τ π h u ∈ (cid:0) Q T ; N , h (cid:1) ( V ) , + ∞ , otherwise . (5.3)5.1. Error estimate of full discretization.
The error u nh − u ( t n ) is divided as u nh − u ( t n ) = (cid:0) u nh − u n (cid:1) + ( u n − u ( t n )) , where the second term in the right-hand side is the error in tempo-ral direction, which has been studied in Proposition 3.1. Hence we only need to consider theerror u nh − u n . By inserting the term π h u n , we get u nh − u n = (cid:0) u nh − π h u n (cid:1) + ( π h u n − u n ) = : e nh + e n π . Note that (4.2) and Propositions 3.3-3.4 yield that, for k ∈ { , } , (cid:0) E k e n π k V (cid:1) ≤ Ch k (cid:16) E k u n k H k ( D ) (cid:17) ≤ Ch k (cid:16) + E k u k D ( M k ) (cid:17) . The estimate of error e nh is stated in the following theorem. Theorem 5.1.
Let { u n , ≤ n ≤ N } in L ( Ω ; H k ( D )) with k ∈ { , } be the solution of (3.1) andlet { u nh , ≤ n ≤ N } in L ( Ω ; V h ) be the solution of (5.1) . Then there is a constant C independentof h and τ such that max ≤ n ≤ N (cid:16) E k e nh k V (cid:17) ≤ Ch k − for k ∈ { , } . (5.4) Proof.
We apply the projection π h to the temporal semidiscretization (3.1) and use Proposition4.1 (i) to get π h u n + = π h u n + τ (cid:0) M h u n + M h u n + (cid:1) − π h ∆ W n + . (5.5)Subtracting (5.1) from (5.5) yields, e n + n = e nh + τ (cid:0) M h e nh + M h e n + h (cid:1) + τ (cid:0) M h e n π + M h e n + π (cid:1) . (5.6)Applying h· , e nh + e n + h i V , we obtain k e n + h k V − k e nh k V = τ h M h ( e nh + e n + h ) , e nh + e n + h i V + τ h M h ( e n π + e n + π ) , e nh + e n + h i V . (5.7)For the second term on the right-hand side of (5.7), we use (4.12) to get h M h ( e n π + e n + π ) , e nh + e n + h i V ≤ − h M h ( e nh + e n + h ) , e nh + e n + h i V + Ch k − k u n + u n + k H k ( D ) . Hence (5.7) becomes k e n + h k V − k e nh k V ≤ τ h M h (cid:0) e nh + e n + h (cid:1) , e nh + e n + h i V + C τ h k − k u n + u n + k H k ( D ) . Proposition 4.1 (ii) leads to h M h e nh + M h e n + h , e nh + e n + h i V ≤ , and then E k e n + h k V − E k e nh k V ≤ C τ h k − E (cid:16) k u n k H k ( D ) + k u n + k H k ( D ) (cid:17) ≤ C τ h k − . Gronwall’s inequality yields the conclusion. (cid:3)
Combining the error estimates in temporal and spatial directions, we finally obtain the errorestimate for the full discretization (5.1).
Theorem 5.2.
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