Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation

DOI:10.1051/m2an/2010100 www.esaim-m2an.org

CONVERGENCE OF A CONSTRAINED FINITE ELEMENT DISCRETIZATION OF THE MAXWELL KLEIN GORDON EQUATION

Snorre H. Christiansen

and Claire Scheid

Abstract. As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.

Mathematics Subject Classification. 65M60, 78M10.

Received February 11, 2009. Revised May 21, 2010.

Published online February 21, 2011.

1. Introduction

Non-linear wave equations are at the heart of basic physical models. Fundamental particles are best described by the quantum version of the Yang-Mills-Higgs equation (YMH) and gravitational fields satisfy the Einstein equation for general relativity (GR). For the former the unknown is a connection on a certain vectorbundle over space-time, whereas for the latter it is a pseudo-Riemannian metric. The equations can in both cases be derived from a variational principle involving a Lagrangian with a large gauge-group giving rise to constraints. Second order hyperbolic partial differential equations involving unknowns from differential geometry and stemming from a variational principle will be calledgeometric wave equations.

The well-posedness of equations with such a rich structure has recently been proved in Sobolev spaces of relatively low regularity. This is relevant both to physics and numerical analysis, since norms related to the energy are the most natural and are most easily incorporated into stability arguments for numerical schemes.

For an introduction to the mathematics of geometric wave equations see [28]. Well posedness in the energy norm for the Yang-Mills equation was proved in [21]. For general relativity, progress on the issue is surveyed in [19].

Numerical models exist for both GR and YMH but little, if any, numerical analysis is available for them.

The only geometric wave equation for which we are aware of convergence proofs is the wave map equation [3].

With the long-term goal of understanding numerical schemes for GR and YMH we propose to study in this

Keywords and phrases. Waves, Maxwell Klein Gordon, non-linear constraints, finite elements, convergence analysis.

∗ This work, conducted as part of the award “Numerical analysis and simulations of geometric wave equations” made under the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) Awards scheme, was supported by funds from the Participating Organizations of EURYI and the EC Sixth Framework Program.

1 CMA c/o Dept. Math, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway. [email protected]

2 Laboratoire Jean Alexandre Dieudonn´e, Universit´e de Nice Sophia Antipolis, 06108 Nice Cedex 02, France.

[email protected]

Article published by EDP Sciences ©EDP Sciences, SMAI 2011

paper the simplest equation in the YMH family, namely the Maxwell-Klein-Gordon (MKG) equation obtained with the gauge group U(1).

In the MKG equation the unknowns are the electromagnetic field, described by a vector potential, and a scalar (complex) field. The scalar field gives rise to a current exciting the electromagnetic field whereas the vector potential enters the coefficients of the wave-equation satisfied by the scalar field. While the wave equation and the Maxwell equations are both linear, the coupling creates a non-linear evolution equation. It is also important that electric charge should be conserved, which gives a non-linear constraint on the flow.

In [10] a finite element method for the YM equation was introduced, whereby the constraint is satisfied through a special application of Lagrange multipliers. In [9] the method was generalized so that it covers all equations in the YMH family and in particular the MKG equation. In this paper we shall prove convergence for this scheme in space dimension 2 with continuous time. The essential features of the scheme are that it preserves energy, which gives control over the curl of the vector potential, and that the constraint preservation gives a weak control over its divergence. Together curl control and divergence control would imply control in the Sobolev space H¹, if it weren’t for the fact that the finite element spaces we use, namely N´ed´elec’s edge elements, are not in this space. Nevertheless we prove a discrete analogue of the Sobolev embedding, valid for N´ed´elec’s edge elements, in the spirit of Kikuchi’s compactness result [18]. The proof uses recently constructed commuting interpolators defined on rough functions. Together with Kato’s inequality, this gives us strong convergence in L^p spaces. A duality argument gives control of the Lagrange multiplier, sufficient to conclude that the limit of the discrete solutions satisfies the continuous MKG equations. The difficulties arising in space dimension 3 are pointed out along the way – the arguments used here fail critically in this case.

The paper is organized as follows. In Section2, after setting up notations, we give some preliminary results.

Since some of the results might be of general interest (in particular the discrete Sobolev embedding), we state and prove them for arbitrary dimension. Section 3 is then dedicated to the exposition of the MKG equation and the semi-discrete scheme we use. Convergence of the approximate solutions of MKG is then obtained in Section 4.

2. Preliminary 2.1. Notations

Fix an integern≥2 and let Ω be a bounded domain in Rⁿ, which is either contractible with a continuously differentiable boundary, or convex. Thus the topology is trivial (curl-free vector fields are gradients) and basic elliptic regularity estimates hold.

2.1.1. Continuous spaces

We recall some standard notations.

L^p spaces:

• L^p(Ω) denotes the classicalL^p space of real valued functions on Ω, forp∈[1,∞].

• We say thatφ∈L^p(Ω,C) if(φ)∈L^p(Ω) and(φ)∈L^p(Ω), whereandare the real and imaginary parts of a complex number. If there is no ambiguity we will sometimes omit theCfrom the notation.

Sobolev spaces:

• H^s(Ω) and H₀^s(Ω), fors∈R, the usual Hilbertian Sobolev spaces of real valued functions. Fors∈N standard norms and seminorms are denoted · _H^s_(Ω) and| · |s. For complex valued functions we write H^s(Ω,C).

• W^s,p(Ω),W₀^s,p(Ω) denote the Banach spaces obtained by generalizing the above spaces toL^pintegrability.

• Recall that fors >0 and 1< p <∞,W^−s,p(Ω) is dual toW₀^s,p(Ω) where ¹_p+_p¹ = 1.

Scalar product:

• We denote by·the canonical scalar product on vectors inRⁿ, and by| · |the associated norm.

• The real-valued scalar product onL²(Ω,C) is denoted·,· , and · is the associatedL²-norm;·,· is also used for the duality products between Sobolev spaces with L²as pivot space.

Spaces of vector fields:

• L²(Ω) is the space of square integrable vector fields and similar definitions hold for H¹(Ω), H^s(Ω), L^p(Ω,C), etc.

• H(curl,Ω) is the space of vector fields inRⁿ with square integrable curl; the analogue space for the divergence will be notedH(div,Ω). For basic results onH(curl,Ω) andH(div,Ω), see [17,25].

• H₀(curl,Ω) := {A∈H(curl,Ω) such thatγ_τA= 0 on∂Ω} where γ_τA is the tangential component ofAon∂Ω.

• V:={v∈H₀(curl,Ω)|divv= 0 in Ω}.

• Forq≥1,H^q₀(curl,Ω) :=H₀(curl,Ω)∩L^q(Ω).

Time dependence. Fix a timeT >0:

• For any closed subinterval I of [0, T] and Banach space X, we letC(I, X) be the space of continuous functions fromItoX, which is a Banach space when equipped with the uniform norm. Also,C(0, T;X) will stand forC([0, T], X).

• Forp∈[1,∞] andX a Banach space,L^p(0, T;X) is the Bochner space defined in [30].

• Cw(0, T;X) denotes the space of functions of time with values in X which are weakly continuous, explicitlyu∈ Cw(0, T;X) means thatt→l(u(t)) is continuous on [0, T], for alllin X, the dual ofX. 2.1.2. Semi-discretization

Let (T_h) be a regular and quasi-uniform family of simplicial meshes of the domain Ω. As usual the parameterh is also the mesh-width ofT_h. We use some standard finite element spaces onT_h, and simplices are usually denoted byK.

• P₁ is the space of affine functions (on some open subset ofRⁿ).

• Y_h⁰ is the space of piecewise affine and continuous real functions on Ω, vanishing on the boundary

∂Ω [12].

• Y_h¹ is the space of N´ed´elec edge element vector fields on Ω (see [4,25]) with vanishing tangential com- ponent on the boundary∂Ω. ThusY¹_h⊂H₀(curl,Ω).

• Z_h⁰ is the space of piecewise affine and continuous complex scalar functions vanishing on the bound- ary∂Ω.

• We also putX_h:=Y_h¹×Z_h⁰.

We define the space of discrete divergence free vectors:

• V_h:=

v_h∈Y¹_h : v_h,gradβ_h = 0, ∀β_h∈Y_h⁰ .

Remark 2.1. Many of the difficulties we encounter are related to the fact that:

V_hV,

but will be resolved by estimating the gap from the former to the latter in appropriate norms.

For the case of a curved boundary we suppose that the mesh and finite element spaces are adapted by appropriate parameterizations, using the techniques in [15].

Throughout the paper, we will use the notation C to refer to a constant independent of h. It might be needed to be taken larger in subsequent steps of our arguments, but we nevertheless keep the same notation throughout.

2.2. Preliminary results

In this section we present some preliminary results. We state them for arbitrary dimension n and will use them in following sections in the particular casen= 2, and in remarks concerningn= 3. These results are either

quite classical, or generalizations of classical results from theL²case to the L^q case, or from time independent fields to time dependent ones.

In the following theorem, whereD denotes the standard gradient, we recall Kato’s inequality.

Theorem 2.2([22]). If A:L²_loc(Rⁿ,Rⁿ),f ∈L²(Rⁿ,C)and(D+ iA)f ∈L²(Rⁿ), then|f|, the modulus of f, is in H¹(Rⁿ)and the diamagnetic inequality:

|D|f|(x)| ≤ |(D+ iA)f(x)|

holds pointwise for almost every x∈Rⁿ.

The second result we state in this section is the well-known Helmholtz decomposition of fields inH(curl,Ω).

We have a statement both in the continuous and in the discrete case. Under the hypotheses for our domain, we havein the continuous case:

For everyu ∈H₀(curl,Ω), there exists a unique˚u ∈Vandp∈H₀¹(Ω) such that:

u =˚u+gradp andin the discrete case:

For everyuh∈Y¹_h, there exists a unique˚uh∈V_h andp_h∈Y_h⁰ such that:

uh=˚uh+gradp_h. These results can be found for example in [2,25].

In the following many results will rely on Sobolev embeddings [1] one variant of which is recalled here:

Proposition 2.3. For all q ∈ ]1, n[, with q defined by 1/q−1/n= 1/q, one has a continuous embedding L^q(Ω)→W^−1,p(Ω),for p≤q. It is compact whenp < q.

The study of the convergence of the scheme relies on norm estimates in both time and space and on the possibility to extract strongly converging subsequences. Thus we need the characterization of compact sets in the time dependent case, in spacesL^∞(0, T;B) whereB is a Banach space; this has been studied for example by Simon in [30]. The following theorem gives a sufficient condition for compactness for subsets ofL^∞(0, T;B).

Theorem 2.4 ([30]). Suppose that X, B, Y are Banach spaces such that X ⊂B ⊂Y with continuous embed- dings, the first being compact. SupposeF is a bounded set inL^∞(0, T;X)such that ^∂F_∂t is bounded inL^r(0, T;Y) for somer >1. ThenF is relatively compact in C(0, T;B).

The next propositions (2.5to2.12) are generalization of some classicalL²results to theL^pcase (Props.2.5,2.8, 2.9and2.12) and/or to the time dependent case (Props.2.7and2.11).

The objective of the next two propositions is to establish an analogue of the usual Kikuchi compactness property, in L^q and to include time dependence. The property is first proved for fields independent of time, then extended to time dependent fields.

Denote by 2^∗the number such that ₂¹∗ =¹₂−_n¹ forn≥2 (with the convention that 2^∗= +∞forn= 2).

Proposition 2.5. Let 1≤q≤2 (q <2 if n= 2). Then there exists C >0such that for all v_h∈V_h, vh_L^q_(Ω)≤Ccurlvh_L²_(Ω).

Furthermore if 1≤q <2^∗,vh∈V_h and curlvh is bounded inL²(Ω), then(out of any subsequence)one can extract a subsequence converging(in norm)in L^q(Ω)to some v∈V.

Proof. Letqbe as in the statement of the theorem. We first prove the bound on theL^q(Ω) norm.

We denote by:

• P theL²-orthogonal projection on the space of square integrable divergence free vectors. The kernel ofP isgradH₀¹and this projection preserves the curl,i.e.

curl◦P =curl. (2.1)

FurthermorePV_h⊂V.

• Q_h the L^p stable projection ontoY_h¹ constructed in [11] (the ones constructed in [2,26] could also be used). It verifies the following property:

Ifcurlv= 0, then curlQ_hv = 0. (2.2)

Forv_h∈Y¹_hwe havePv_h∈Vand curlPv_h∈L^q(Ω) soPv_h∈W^1,q(Ω) (using arguments of regularity of solution of elliptic problems, see for example [29]). We get also the estimate:

|Pvh|1,q≤CcurlPvh_L^q_(Ω). (2.3) Remark 2.6. We havecurlv_h=curlPv_h=curlQ_hPv_hforv_h∈Y_h¹.

Choose nowv_h∈V_h. By triangular inequality,

v_hL^q_(Ω)≤ v_h−Q_hPv_hL^q_(Ω)+Q_hPv_h−Pv_hL^q_(Ω)+Pv_hL^q_(Ω). (a) We have by Bramble-Hilbert type estimates:

Pv_h−Q_hPv_hL^q_(Ω)≤Ch|Pv_h|_1,q ≤ChcurlPv_hL^q_(Ω)≤Chcurlv_hL^q_(Ω), so that we can use the inverse inequality (4.5.11) in [6] and obtain:

Pv_h−Q_hPv_hL^q_(Ω)≤Chh^{min(0,nq−n2)}curlv_hL²_(Ω). (2.4) (b) Furthermore

v_h−Q_hPv_hL^q_(Ω)≤Ch^{min(0,nq−n2)}v_h−Q_hPv_hL²_(Ω).

But sincecurl(vh−Q_hPvh) = 0, we have that vh⊥vh−Q_hPvh, andPvh⊥vh−Q_hPvh. So that v_h−Q_hPv_h²_L2(Ω)≤ v_h−Q_hPv_hL²_(Ω)Pv_h−Q_hPv_hL²_(Ω).

Therefore

v_h−Q_hPv_hL^q_(Ω)≤Ch^{min(0,nq−n2)}Pv_h−Q_hPv_hL²_(Ω). By (2.4), one concludes that

vh−Q_hPvh_L^q_(Ω)≤Chh^{min(0,nq−n2)}curlvh_L²_(Ω). (2.5)

(c) By the Sobolev embedding ofH¹(Ω) intoL^q(Ω), we have:

Pv_hL^q_(Ω)≤CPv_hH¹_(Ω). Friedrich’s inequality yields then

Pv_hL^q_(Ω)≤Ccurlv_hL²_(Ω). (2.6) (d) Since 1 +ⁿ_q −ⁿ₂ ≥0, we combine (2.4)–(2.6) to conclude that for someC >0:

∀vh∈V_h, vh_L^q_(Ω)≤Ccurlvh_L²_(Ω). (2.7) It remains to prove that a subsequence ofvhconverges strongly in L^q(Ω), if 1≤q <2^∗.

Since (Pvh) is bounded inH¹(Ω) we deduce strong convergence inL^q(Ω) after subsequence extraction. Then since for 1≤q <2^∗ we have 1 +n/q−n/2>0, from (2.4) and (2.5) we deduce that (vh) converges inL^q(Ω)

and has the same limit asPvh inL^q(Ω). This concludes the proof.

Proposition2.5 can be generalized to fields with a time dependence:

Proposition 2.7. Let 1< q≤2^∗ (q <2^∗ ifn= 2).

There existsC >0 such that for allvh∈L^∞(0, T;V_h)

v_hL^∞_(0,T;L^q_(Ω))≤Ccurlv_hL^∞_(0,T;L²_(Ω)). Furthermore if there existsC >0 such that

curlv_hL^∞_(0,T;L²_(Ω))≤C, and

v˙_hL^∞_(0,T;L²_(Ω)) ≤C,

then for allq <2^∗there existsv ∈L^∞(0, T;V)such that a subsequence of(v_h)converges in theL^∞(0, T;L^q(Ω)) norm tov.

Proof. As all inequalities from the proof of Proposition 2.5 can be transported to time dependent fields, the only point which has to be clarified is that a subsequence ofvh has a limit inL^∞(0, T;L^q(Ω)).

Remark that if (vh) is bounded inL^∞(0, T;H(curl,Ω)) then (Pvh) is bounded inL^∞(0, T;H¹(Ω)). More- over,P commutes with time-differentiation and therefore

P˙v_hL^∞_(0,T;L²_(Ω))≤ v˙_hL^∞_(0,T;L²_(Ω)) ≤C, wherePv˙hdenotes the time derivative of Pvh.

Applying Theorem 2.4, Pvh converges strongly (considering a subsequence) in L^∞(0, T;L²(Ω)). Using inequalities (2.4) and (2.5) for time dependent fields,vh−PvhL^∞(0,T;L²(Ω)) converges to 0 as htends to 0.

Then an interpolation inequality between L^p spaces, the convergence of vh in L^∞(0, T;L²(Ω)), and the fact

that v_hL^∞_(0,T;L^q_(Ω)) is bounded, complete the proof.

The next proposition gives stability results for projections onto finite element spaces. This result will be needed in Section4.5.

Proposition 2.8. Let P_h¹ be theL² projection on Y¹_h andP_h⁰ be theL² projection on Z_h⁰. Then:

(a) P_h¹ is stable inL^p, and fromH¹(Ω)toH(curl,Ω).

(b) P_h⁰ is stable inH^s(Ω), for all−1≤s≤1.

Proof. (a) ForP_h¹:

• Stability inL^p(Ω)

Using the result in [14] in the vectorial case, one obtains stability in theL^p-norm.

• Stability inH(curl,Ω)

LetQ_hbe the operator constructed in [11] already used in the proof of Proposition2.5. It is stable both inL²(Ω) and H(curl,Ω). Using also the inverse inequality betweenH(curl,Ω) andL²(Ω), one has:

∀u ∈H¹(Ω),P_h¹u_H(curl,Ω)≤ Q_hP_h¹u−Q_hu_H(curl,Ω)+Q_hu_H(curl,Ω) (2.8)

≤Ch⁻¹Q_hP_h¹u−Q_hu_L²_(Ω)+Q_hu_H(curl,Ω) (2.9)

≤Ch⁻¹P_h¹u−u_L²_(Ω)+Q_hu_H(curl,Ω) (2.10)

≤Cu_H¹_(Ω). (2.11)

(b) ForP_h⁰:

• The stability inH¹(Ω) comes from the result in [5,13].

• The stability inH⁻¹(Ω) follows by duality.

• The stability inH^s(Ω), for−1≤s≤1 is obtained by using interpolation inequalities.

Discretizing continuous equations leads to discrete ones which should have good convergence properties. The next proposition states this for a particular class of equations.

Proposition 2.9. Let p∈]1,+∞[ be given,a(·,·)the bilinear form on W^1,p(Ω)×W^1,p(Ω) (1/p+ 1/p = 1) given by:

a(u, v) =

Ω

gradu·gradv, and for h >0,f_h∈W^−1,p(Ω),f ∈W^−1,p(Ω).

Let also u_h∈Y_h⁰ be the solution of

a(u_h, v_h) =f_h, v_h , ∀v_h∈Y_h⁰, andu∈W₀^1,p(Ω)the solution of

a(u, v) =f, v , ∀v∈W₀^1,p(Ω).

Then:

(i) u_hW^1,p_(Ω)≤Cf_hW^−1,p_(Ω). (ii) If f_h−→

h→0f in W^−1,p(Ω), thenu_h−→

h→0uinW₀^1,p(Ω).

This is essentially a reformulation of the following fact:

Remark 2.10. The bilinear formaverifies a uniform discrete inf-sup condition in the normsW^1,p(Ω)×W^1,p(Ω) see [6,29].

The following proposition is a generalization to time dependent fields:

Proposition 2.11. Let T > 0 and p ∈ ]1,+∞[ be given, a(·,·) the bilinear form on W^1,p(Ω)×W^1,p(Ω) (1/p+ 1/p = 1)given by:

a(u, v) =

Ω

gradu·gradv, and for h >0,f_h∈L^∞(0, T;W^−1,p(Ω)),f ∈L^∞(0, T;W^−1,p(Ω)).

Let also u_h∈L^∞(0, T;Y_h⁰)be the solution of

a(u_h(t), v_h) =f_h(t), v_h , ∀v_h∈Y_h⁰, for a.e. t in[0, T],

andu∈L^∞(0, T;W₀^1,p(Ω))the solution of

a(u(t), v) =f(t), v , ∀v∈W₀^1,p(Ω) for a.e. t in[0, T].

Then:

(i) u_hL^∞_(0,T;W^1,p_(Ω)) ≤Cf_hL^∞_(0,T;W^−1,p_(Ω)). (ii) If f_h−→

h→0f in L^∞(0, T;W^−1,p(Ω)), thenu_h−→

h→0uinL^∞(0, T;W₀^1,p(Ω)).

We will also needL^p stability of the Helmholtz decomposition as stated in the following:

Proposition 2.12. The discrete Helmholtz decomposition inY¹_h is stable inL^p-norm.

Proof. Let E_h ∈ Y_h¹ and let ˚E_h ∈ V_h and p_h ∈ Y_h⁰ satisfy E_h = ˚E_h+gradp_h. Keeping notations from Proposition2.9, we deduce that divE_h∈W^−1,p(Ω), anda(p_h, v_h) =divE_h, v_h for allv_h∈Y_h⁰. Therefore:

p_hW^1,p_(Ω)≤CE_hL^p_(Ω).

Stability of the decomposition follows.

Finally we state a result on compact perturbations (Prop.2.13) and a result on dual estimates (Prop.2.14) which we will use in Section 4.5 to get estimates on the time derivative of the discrete solutions and on the Lagrange multiplier.

The following proposition is a generalization of the result obtained in [7], Proposition A.5.2 (see also [8]

Thm. 1.12, Cor. 1.17). The dual space of a Banach spaceY is denotedY.

Proposition 2.13. Let X and Y be two reflexive Banach spaces and A : X →Y^∗ a continuous linear map with closed range. Let K denote a relatively compact set of compact operators X →Y^∗. Let(X_h)and(Y_h)be two families of finite-dimensional subspaces ofX andY. Suppose that(Y_h)verifies an approximation property:

∀y∈Y, lim

h→0 inf

y∈Yhy−y= 0,

that A satisfies a discrete uniform inf-sup condition onX_h×Y_h, and for allB ∈ K,A+B is injective. Then there exists a constant C such that for all B ∈ K, A+B verifies a uniform discrete inf-sup condition with constant C.

Proof. We apply Proposition A.5.2 in [7]. For everyB ∈ K, one can construct a ballB(B, r_B) of centerB and radiusr_Bsufficiently small, such that forB∈B(B, r_B),A+B verifies an inf-sup condition independent ofB. Denote the corresponding constant by C_B. Since {B(B, r_B),B ∈ K} coversK, we can extract from it a finite subcover. LetCbe the worst inf-sup constant in this finite family. Then for allB ∈ K,A+Bverifies a uniform

inf-sup condition with constantC. This concludes the proof.

Proposition 2.14. Let X andY be two Banach spaces equipped with respectively the norms · _X and · _Y, a(·,·)a continuous bilinear form onX × Y.

Then let (X_h)and (Y_h) be two families of subspaces of equal finite dimension ofX andY respectively. We suppose that a(·,·)verifies a discrete inf-sup condition on Xh× Yh. We consider T_h :Y → Xh, such that for all u∈ Y:

a(T_hu, v_h) =u, v_h , ∀v_h∈ Yh, (2.12) andT_h :X→ Yh, such that for all v∈ X:

a(u_h, T_hv) =u_h, v , ∀u_h∈ Xh. (2.13)

Let X₊ and Y₋ be two other Banach spaces (with respective norms ._X+ and ._Y−) such that X ⊂ X₊ and Y−⊂ Y, and suppose that if v∈ X₊ thenT_hv∈ Y− and one has

T_hvY− ≤Cv_X+. (2.14)

Then for alluin Y₋ ,

T_hu_X+≤Cu_Y− .

Proof. Existence of solutions is guaranteed by inf-sup conditions. From (2.12)–(2.14), we deduce:

T_hu_X+= sup

v∈X₊

T_hu, v

v_X+ = sup

v∈X₊

a(T_hu, T_hv) v_X+ = sup

v∈X₊

u, T_hv

v_X+ ≤Cu_Y− . This proposition generalizes to the time dependent case in an obvious way.

These preliminary results are valid for arbitrary dimension n. From now on we will consider a domain Ω included inR², and study the Maxwell Klein Gordon equation in this case. However all along the article the difficulties in dimensionn= 3 will be pointed out.

3. Equation and discrete formulation

Letn= 2 so that Ω⊂R².

3.1. Continuous formulation

LetT >0 be given. Solving the Maxwell Klein Gordon equation consists in finding:

• a time dependent gauge potential defined on [0, T],t→ A(t) =

α(t) A(t)

whereα(t) is a real function on Ω andA(t) a real vector field on Ω; and

• a time dependent complex scalar function on Ω, defined on [0, T]: t→φ(t), which constitute a critical point of theaction:

S(A, φ, α) =1 2

_T

0 curl A²_L2(Ω)− gradα−A˙²_L2(Ω)+D_Aφ²_L2(Ω)− φ˙+iαφ²_L2(Ω).

As before D denotes the spatial gradient operator acting on complex functions and D_Aφ =Dφ+ iAφ is the covariant derivative ofφ.

We can express the variation ofSat (α,A, φ) in the direction (α,A, φ). Then the stationarity of the action gives Euler-Lagrange equations:

• variation with respect toA gives an evolution equation for A,

• variation with respect toφ gives an evolution equation forφ,

• variation with respect toα gives a constraint on the flow.

For more details on this we refer to [9].

3.1.2. In the temporal gauge

From now on we turn to the Maxwell Klein Gordon equation in temporal gauge, that is, we imposeα(t)≡0.

The equations to solve on [0, T] are then:

A˙ = −E, (3.1)

φ˙ = −ψ, (3.2)

E˙ = curl(curl(A)) +(DAφφ),¯ (3.3)

ψ˙ = D^∗_AD_Aφ. (3.4)

The constraint is given by:

div(E) +(ψφ) = 0.¯ (3.5)

We suppose that initial conditions are:

A(0, .) = A⁰(.)∈H₀(curl,Ω)∩H¹(Ω), (3.6)

φ(0, .) = φ⁰(.)∈H₀¹(Ω,C), (3.7)

E(0, .) = E⁰(.)∈L²(Ω), (3.8)

ψ(0, .) = ψ⁰(.)∈L²(Ω,C), (3.9)

and that they verify the constraint given by (3.5) (inH⁻¹(Ω)).

We define the energy of the field at timetby:

H(t) =E,E +curl A,curl A +ψ, ψ +D_Aφ, D_Aφ, (3.10) and have that

H(0)<+∞. (3.11)

Proposition 3.1. This energy is conserved in time for smooth solutions.

In the rest of the paper we often drop the complex signCfor simplicity of notation.

Weak solution. We introduce the notion of weak solution to (3.1)–(3.4).

Definition 3.2. (E,A, ψ, φ) is said to be aweak solutionof (3.1)–(3.4), if

• We have:

– E∈ C(0, T;H⁻¹(Ω))∩L^∞(0, T;L²(Ω)),

– A∈ C(0, T;L²(Ω))∩L^∞(0, T;H₀(curl,Ω)∩W^1,q(Ω)) forq <2, – ψ∈ C(0, T;H⁻¹(Ω))∩L^∞(0, T;L²(Ω)),

– φ∈ C(0, T;L²(Ω))∩L^∞(0, T;H₀¹(Ω)).

•

A˙ = −E, φ˙ = −ψ.

• For every (E, ψ)∈ C_c^∞(]0, T[×Ω)²× C_c^∞(]0, T[×Ω), there holds

− _T

0 E,E˙ dt− _T

0 ψ,ψ˙ dt= _T

0 curl A,curl E dt+ _T

0 D_Aφ,iφE dt+ _T

0 D_Aφ, D_Aψ dt. (3.12)

3.2. Discrete formulation

Considering the variational formulation of (3.3) and (3.4), and simply discretizing in space inX_h provides a scheme which violates the constraint (3.5). In order to preserve it we consider the following constraint preserving scheme proposed in [9] and formulated as a saddle point problem:

ForT >0, findt→(A_h(t), φ_h(t))∈X_hand a Lagrange multipliert→β_h(t)∈Y_h⁰such that for allt∈[0, T]:

A˙_h = −E_h, (3.13)

φ˙_h = −ψ_h, (3.14)

and for all (E_h, ψ_h)∈X_h and allβ_h ∈Y_h⁰:

E˙_h,E_h +ψ˙_h, ψ_h +E_h,gradβ_h − ψ_h,iφ_hβ_h = curl A_h,curl E_h +D_Ahφ_h,iφ_hE_h +D_Ahφ_h, D_Ahψ_h , (3.15)

E˙_h,gradβ_h − ψ˙_h, iφ_hβ_h = 0, (3.16)

with initial conditions:

A_h(0, .) = A⁰_h∈Y_h¹, (3.17)

E_h(0, .) = E⁰_h∈Y¹_h, (3.18)

φ_h(0, .) = φ⁰_h∈Z_h⁰, (3.19)

ψ_h(0, .) = ψ_h⁰∈Z_h⁰, (3.20)

where we suppose thatA⁰_h,E⁰_h,φ⁰_h,ψ⁰_hare chosen such that:

(i) A⁰_h−→

h→0A⁰ inH^q₀(curl,Ω), ∀q <+∞(forn≥3 replace byq≤2), (ii) E⁰_h−→

h→0E⁰in L²(Ω), (iii) φ⁰_h−→

h→0φ⁰ inH¹(Ω), (iv) ψ⁰_h−→

h→0ψ⁰ inL²(Ω),

and the following constraint is satisfied initially:

E⁰_h,gradβ_h − ψ_h⁰,iφ⁰_hβ_h = 0, ∀β_h ∈Y_h⁰. (3.21) The rationale behind the discrete constraint is as follows. Differentiating (3.5) in time gives:

div( ˙E) +( ˙ψφ¯+ψφ) = 0.˙¯ (3.22)

Then we remark that for solutions:

(ψφ) =˙¯ −(ψψ) = 0.¯ (3.23)

Finally we remark that fora∈Candb∈Rwe have:

(a)b=(¯aib) (3.24)

so that testing the constraint with a realβ leads to (3.16).

Notice that it is possible to choose initial conditions satisfying the given constraint. Indeed, one choosesφ⁰_h, ψ⁰_h andA⁰_hsuch that (iii), (iv) and (i) are verified. One then considers the Helmholtz decomposition ofE⁰:

E⁰=˚E⁰+gradp⁰.

Then definingp⁰_h as the solution ofgradp⁰_h,gradβ_h − ψ_h⁰,iφ⁰_hβ_h = 0 for all β_h ∈Y_h⁰, one concludes using results from Section 2.2 that p⁰_h converges top⁰ in H¹(Ω). Choosing˚E⁰_h such that˚E⁰_h −→

h→0

˚E⁰ in L²(Ω), and defining E⁰_h:=˚E⁰_h+gradp⁰_hgives the desired property.

3.2.2. Existence of a solution to the discrete formulation

The above equation can be viewed with (A_h, φ_h) ∈ X_h as given parameters. We then can rewrite equa- tions (3.15) and (3.16) as:

E˙_h,E_h +ψ˙_h, ψ_h +E_h,gradβ_h − ψ_h,iφ_hβ_h = f_Ah,φh(E_h) +g_Ah,φh(ψ_h), (3.25) E˙_h,gradβ_h − ψ˙_h,iφ_hβ_h = 0, (3.26) where

f_Ah,φh(E_h) =curl A_h,curl E_h +D_Ahφ_h,iφ_hE_h and

g_Ah,φh(ψ_h) =D_Ahφ_h, D_Ahψ_h .

Proposition 3.3. Let h > 0 be fixed. The system given by(3.25)and (3.26) with unknowns( ˙E_h,ψ˙_h, β_h) has a unique solution in X_h×Y_h⁰. Furthermore, the solution depends smoothly on the parameters(A_h, φ_h).

TakingE_h=gradβ_h andψ_h= 0 gives the following discrete Babuska-Brezzi compatibility condition:

β_hinf∈Y_h⁰ sup

(E_h,ψ_h)∈Xh

E_h,gradβ_h − ψ_h,iφ_hβ_h

(|β_h|²+|gradβ_h|²)¹²(|E_h|²+|ψ_h|²)¹² ≥ 1 C,

whereC is a positive constant independent of the timetand ofφ_h. Sincehis fixed and all spaces we are dealing with are of finite dimension and all the considered operators are polynomial in the unknowns, we have proved

the proposition.

We denoteP_Y¹

h the projection fromX_h×Y_h⁰onY¹_h, andP_Z⁰

h the projection fromX_h×Y_h⁰onZ_h⁰. IfS is the solution operator associated to equation (3.25), we are able to solve inX_hthe equations

A¨_h=−P_Y¹_h◦ S(A_h, φ_h), (3.27) φ¨_h =−P_Zh⁰◦ S(Ah, φ_h), (3.28) locally in time with initial conditions given by (3.6)–(3.9).

This implies that we have existence of (A_h, φ_h)∈X_hfor the discrete formulation locally in time.

We define the discrete energy at any time:

Hh(t) = 1

2(E_h,E_h (t) +curl A_h,curl A_h (t) +ψ_h, ψ_h (t) +D_Ahφ_h, D_Ahφ_h (t)).

The constraint associated with the discrete formulation of (3.1)–(3.4) is given by:

E_h,gradβ_h − ψ_h,iφ_hβ_h = 0, ∀β_h ∈Y_h⁰. (3.29) One can find in [9] a detailed proof of the following:

Proposition 3.4. Equations (3.13)–(3.16)preserve the constraint (3.29)and the energy of the system.

Energy conservation guarantees that the solutions of equations (3.13)–(3.20) are defined on the whole time- interval [0, T].

We now would like to prove that the sequence (E_h,A_h, ψ_h, φ_h) converges (in a sense which has to be made precise) to a weak solution of the Maxwell Klein Gordon equation (in the sense of Def.3.2).