SMAI-JCM SMAI Journal of Computational Mathematics

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1 SMAI-JCM SMAI Journal of Computational Matematics Compatible Maxwell solvers wit particles II: conforming and non-conforming 2D scemes wit a strong Faraday law Martin Campos Pinto & Eric Sonnendrücker Volume 3 (2017), p <ttp://smai-jcm.cedram.org/item?id=smai-jcm_ _0> Société de Matématiques Appliquées et Industrielles, 2017 Certains droits réservés. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de matématiques ttp://

2 SMAI Journal of Computational Matematics Vol. 3, (2017) Compatible Maxwell solvers wit particles II: conforming and non-conforming 2D scemes wit a strong Faraday law Martin Campos Pinto 1 Eric Sonnendrücker 2 1 CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France address: campos@ljll.mat.upmc.fr 2 Max-Planck Institute for plasma pysics, Boltzmannstr. 2, D Garcing, Germany, Matematics Center, TU Munic, Boltzmannstr. 3, D Garcing, Germany address: sonnen@ipp.mpg.de. Abstract. Tis article is te second of a series were we develop and analyze structure-preserving finite element discretizations for te time-dependent 2D Maxwell system wit long-time stability properties, and propose a carge-conserving deposition sceme to extend te stability properties in te case were te current source is provided by a particle metod. Te scemes proposed ere derive from a previous study were a generalized commuting diagram was identified as an abstract compatibility criterion in te design of stable scemes for te Maxwell system alone, and applied to build a series of conforming and non-conforming scemes in te 3D case. Here te teory is extended to account for approximate sources, and specific cargeconserving scemes are provided for te 2D case. In tis second article we study two scemes wic include a strong discretization of te Faraday law. Te first one is based on a standard conforming mixed finite element discretization and te long-time stability is ensured by te natural L 2 projection for te current, also standard. Te second one is a new non-conforming variant were te numerical fields are sougt in fully discontinuous spaces. In tis 2D setting it is sown tat te associated discrete curl operator coincides wit tat of a classical DG formulation wit centered fluxes, and our analysis sows tat a non-standard current approximation operator must be used to yield a carge-conserving sceme wit long-time stability properties, wile retaining te local nature of L 2 projections in discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are ten provided to validate te stability of te proposed metods. Mat. classification. 35Q61, 65M12, 65M60, 65M75. Keywords. Maxwell equations, Gauss laws, structure-preserving, PIC, carge-conserving current deposition, conforming finite elements, discontinuous Galerkin, Conga metod. 1. Introduction Like its companion article [10], tis work addresses te issue of long-time stability in time-dependent Maxwell solvers, eiter considered alone or coupled wit an additional sceme for te current sources. It is known tat tis issue is strongly related to te good preservation of te divergence constraints at te discrete level. We refer to te introduction of [10] for a review of te literature on wic our work is based, and for a presentation of its main guiding lines. In tis article we pursue tis study and propose two compatible scemes tat include discrete Faraday laws in strong form for te 2D Maxwell system t B + curl E = 0 t E c 2 curl B = 1 (1.1) J. ε 0 Te first sceme is a standard curl-conforming mixed finite element metod for wic we verify tat a standard Galerkin (L 2 ) projection provides a Gauss-compatible approximation for te current J, 91

3 M. Campos Pinto & E. Sonnendrücker in te sense of [11]. Te second sceme extends tis construction to spaces of fully discontinuous fields as in standard DG metods, in order to avoid inverting global mass matrices. It belongs to te class of conforming/non-conforming (Conga) metods designed in [11] to preserve te mixed form of conforming Galerkin approximations, and it also comes wit a compatible approximation operator for te current. An interesting feature of tis particular metod is tat, in 2D, te associated discrete curl operator coincides wit tat of a classical DG metod wit centered fluxes. However, te current approximation operator is non-standard. We ten complete tese two scemes by identifying for eac of tem te discrete divergence operators tat form a complete structure-preserving discretizations in te sense of [10]. In te case were te Maxwell system (1.1) is coupled wit an additional equation for te source, suc as a Vlasov equation t f + v x f + q m (E + v B) vf = 0 (1.2) describing te collisionless evolution of one or several particle species wit carge q and mass m troug teir distribution function f = f(t, x, v), tis framework allows us to sow tat te approximation operators identified as Gauss-compatible for te Maxwell system alone can also be used to deposit te associated current density ˆ J := vf dv (1.3) (or better, its approximation by numerical particles) on te finite element spaces in a stable, cargeconserving way, be it for te conforming or te non-conforming Galerkin discretization. Finally, we provide numerical experiments tat validate te approac and te numerical convergence of te proposed scemes (establised by teoretical means for te Maxwell system alone), using a pure Maxwell problem and an academic Maxwell-Vlasov test case. Te outline is as follows: In Section 2 we briefly recall te main criterion identified by te stability analysis developped in our previous works [11, 10] to te case considered ere of a 2D Maxwell equations wit a strong Faraday law. Ten in Section 3 we introduce te needed discrete function spaces for a conforming Finite Element approximation and verify tat tey fit into our abstract framework. We next consider in Section 4 te case of discontinuous non conforming Finite Elements were our framework enables in a non trivial way to construct long time stable discretizations. We also sow tat in 2D, tis new sceme can be formulated as a Discontinuous Galerkin (DG) sceme wit a non-standard current approximation metod. In Section 5 we ten sow ow to construct an approximation of te current from te particles tat yields a stable sceme, wic in te DG case can be seen as a standard deposition metod wit a local correction. All tis is finally validated in Section 6 on two simple but relevant test cases. 2. Teoretical framework To analyze te conforming and non-conforming metods described in te following sections we rely on te tools provided in Section 2 of our companion article [10]. Tere te 2D Maxwell equations on a bounded domain Ω were reformulated using a sequence of operators V 0 d 0 =ι V 1 d 1 =curl V 2 d 2 =div V 3 0 {0} (2.1) wit ι te canonical injection from R to L 2 (Ω), and respective domains V 0 = R, V 1 = H(curl, Ω), V 2 = H(div, Ω) and V 3 = L 2 (Ω). If Ω is a bounded and simply-connected Lipscitz domain ten te sequence (2.1) is exact, and so is te dual sequence of adjoint operators {0} 0 V 3 (d 2 ) = grad V 2 (d 1 ) =curl V 1 (d 1 ) =curl V 0 (2.2) 92

4 Compatible Maxwell solvers wit particles, II wit domains V3 = H1 0 (Ω), V 2 = H 0(curl, Ω), V1 = L2 (Ω) and V0 = R. Letting ten A be defined by ( 0 (d A = c 1 ) ) ( ) 0 curl d 1 = c on V = V 0 curl 0 1 V2 = H(curl, Ω) H 0 (curl, Ω), te time-dependent Maxwell equations (1.1) wit metallic boundary conditions can be rewritten as t U AU = F (2.3) wit U = (cb, E) T, F = (0, ε 1 0 J)T, and te Gauss laws in te reduced 2D setting can be recast as DU = R (2.4) were R := ( ffl Ω cb0, ε 1 0 ρ)t represents te carge density in tis 2D model, and D is a composite divergence operator defined by D = ( (d 0 ) 0 0 d 2 ) = (ffl ) Ω 0 0 div on V 1 V 2 = L 2 (Ω) H(div, Ω). For completeness we recall te following definitions from Section 2 of [10], as tey are central in our stability and error analysis. First a notion of Gauss-compatible approximation was derived from our previous work [11], wic allows for long-stable scemes in te case of exact sources (see e.g. Corollaries 2.6 and 2.7 in [10]). Definition 2.1 (Def. 2.3 from [10]). We say tat a discrete operator A : V V forms a Gausscompatible approximation of A togeter wit a mapping Π on V if tere exists an auxiliary mapping ˆΠ : ˆV V tat converges pointwise to te identity as 0, and tat is suc tat olds on ˆV. Π A = A ˆΠ (2.5) Ten, notions of structure-preserving and carge-conserving discretizations were introduced to guarantee long-time stability estimates in te case of approximate sources as detailed in Sections 2.1 and 2.4 of [10], see in particular Teorem Since in tis article we design scemes based on te second sequence (2.2), te appropriate definition is as follows. Definition 2.2 (Def and Lemma 2.13 from [10]). We say tat a semi-discrete 2D Maxwell system of te form t B + curl E = 0 t E c 2 curl B = 1 ε 0 J wit curl : V 1 V 2 curl := (curl ) : V 2 V 1 (2.6) completed wit discrete Gauss laws of te form div E = 1 ρ ε 0 (ι ) B = (ι ) B 0 wit is structure-preserving if te following properties old. div : V 2 V 3 ι : V 0 V 1 (2.7) Exact sequence property: wit grad := (div ), te sequence V 3 grad V 2 curl V 1 (ι ) V 0 (2.8) is exact, in te sense tat grad V 3 = ker curl and curl V 2 = ker(ι ). 93

5 M. Campos Pinto & E. Sonnendrücker Stability: te operators in te above sequence satisfy Poincaré estimates, u c P grad u, u V 3 (ker grad) u c P curl u, u V 2 (ker curl ) u c P (ι ) u, u V 1 (ker(ι ) ). wit a constant c P independent of. (2.9) wit In Lemma 2.14 from [10] it is observed tat if te discrete system (2.6) is put under te form { t U A U = F D U = R (2.10) A := c ( ) ( 0 curl : (V curl 0 1 V 2 ) (V 1 V 2 (ι ) ), D := ) 0 : (V 0 div 1 V 2 ) (V 0 V 3 ) ten properties (2.8)-(2.9) old if and only if te composite curl and divergence operators satisfy: (i) Z c P A Z, Z (ker A ) (unif. stability of A ) (2.11) (ii) Z c P D Z, Z (ker D ) (unif. stability of D ) (2.12) (iii) ker D = (ker A ) (compatibility of te kernels). (2.13) Te purpose of Definition 2.2 is to guarantee te long-time stability of te solutions to te full discrete Maxwell system (2.10). A criterion on te discrete sources is ten introduced to guarantee tat solutions to te discrete Ampère and Faraday equations also satisfy te proper discrete Gauss law. Definition 2.3 (Def from [10]). We say tat a semi-discrete Maxwell system of te form (2.10) wit A a skew-symmetric operator, is carge-conserving if (i) it is structure-preserving in te sense tat properties (2.11), (2.12) and (2.13) old, (ii) and te approximate sources satisfy te corresponding discrete continuity equation t R + D F = 0. (2.14) Finally we observe tat wit te notation of Definition 2.2, te first equation from (2.14) is trivial and te second one rewrites as t ρ + div J = 0. (2.15) 3. Conforming elements for te 2D Maxwell system wit a strong Faraday law Altoug it makes no difference on te continuous problem weter one takes te sequence (2.1) or its dual version (2.2) for te primal complex, on te discrete level it leads to two different types of Galerkin metods. In our companion article [10] we ave described te first coice wic leads to a strong discretization of te Ampère equation wit natural boundary conditions. In tis article we consider te second coice wic leads to a strong discrete Faraday equation wit essential boundary conditions. Because te construction of our new non-conforming metod relies on a good understanding of te conforming tools, we now verify tat te structure of te Finite Element exterior calculus introduced by Arnold, Falk and Winter [1, 2, 3], linking te conforming Galerkin approximations of te different Hilbert spaces perfectly fits in te framework introduced in te previous sections. Specifically, tis will allow us to sow in Section 3.4 tat a standard sequence of Finite Element spaces can be equipped 94

6 Compatible Maxwell solvers wit particles, II wit a Gauss-compatible approximation operator in te sense of Definition 2.1, and in Section 5.1 we will verify tat it naturally yields a structure-preserving discretization of Maxwell s equations in te sense of Definition 2.2. Our non-conforming approximation will consist in giving more freedom in te coice of te discrete spaces, in particular to include discontinuous broken spaces, by carefully coosing te projection operators and discrete differential operators so as to preserve te compatibility and structure-preserving properties Mes notations For te mes elements we use te same notations as in [10]. In particular, we assume tat te domain Ω is partitioned by a regular family of conforming simplicial meses T wit maximal triangle diameter tending to zero. We denote by E(T ) te edges of a triangle T T, and E := T T E(T ) te set of all te edges in te mes. Boundary edges are stored in E B. Assuming tat te triangles in T are given arbitrary indices i = 0,..., #(T ) 1, we fix an orientation for te edges as follows. Given e E, we let T (e) be te triangle of minimum index for wic e is an edge, and if e is not a boundary edge we denote by T + (e) te oter triangle saring e. Te edge e is ten oriented by setting n e := n T (e) e, were n T e denotes te outward unit vector of T tat is normal to e, for any e E(T ) Conforming Finite Elements wit a strong Faraday law To derive finite element scemes wit a strong Faraday equation we approximate te non-trivial spaces in te dual sequence (2.2), i.e., V 3 = H1 0 (Ω) (d 2 ) = grad V 2 = H 0(curl; Ω) (d 1 ) =curl V 1 = L2 (Ω) (3.1) by a sequence of discrete spaces V 3 (d 2 ) V 2 (d 1 ) V 1. In tis section we opt for a conforming sequence, i.e., suc tat V 3 H0 1 (Ω), V 2 H 0 (curl; Ω), V 1 L 2 (Ω), (3.2) so tat (d l ) can be defined as te restriction of (d l ) to V l+1. Observe tat ere we denote standard (strong) differential operators as dual ones, and conversely te plain notation d l : V l V l+1 will be used to denote te discrete adjoint of (d l ), wic only makes sense in a weak form, using test functions from V l+1. Tis rater unusual coice is motivated by our desire to use notations consistent wit our companion article [10] were te discretization is performed on te sequence (2.1) wic is seen as te primal one, but obviously te oter coice could be made as well by calling (2.2) te primal sequence. Again, several options are possible for te conforming spaces in (3.2). Here we focus on a standard strategy (see, e.g., [20]) were te spaces V l are respectively defined as a continuous Finite Element space, a Nédélec Finite Element space of te first kind and a discontinuous Galerkin Finite Element space, but we note tat oter coices are possible, see e.g. Remark 4.7 below. Specifically, given an integer degree p 1 we take were V 3 := L p,0(ω, T ) (d 2 ) := grad V 3 V 2 := N p 1,0(Ω, T ) (d 1 ) :=curl V 2 V 1 := P p 1(T ) (3.3) P p 1 (T ) := {v L 2 (Ω) : v T P p 1 (T ), T T } (3.4) 95

7 M. Campos Pinto & E. Sonnendrücker denotes te space of piecewise polynomials of maximal degree p 1 on te triangulation T, L p,0 (Ω, T ) := P p (T ) C 0 (Ω) (3.5) corresponds to te continuous Lagrange elements wit omogeneous boundary conditions, and N p 1,0 (Ω, T ) := N p 1 (T ) H 0 (curl; Ω) wit N p 1 (T ) := P p 1 (T ) 2 + ( y ) P x p 1 (T ) (3.6) is te (first-kind) Nédélec Finite Element space of order p 1 (tus of maximal degree p), again wit omogeneous boundary conditions, see e.g., [4]. Remark 3.1. To be conforming in H(curl; Ω), te piecewise polynomial space V 2 must be composed of vector fields tat ave no tangential discontinuities on te edges of te mes, in te sense tat for any u V 2 te tangential trace n e u on an edge e must be te same wen defined eiter from T (e) or from T + (e). In particular, every basis of V 2 must contain some vector fields supported on two adjacent cells at least. For te sake of completeness we recall te following well-kown result wic will be central to our analysis. Te proof is almost te same as te one of Lemma 3.2 in [10], up to te boundary conditions wic are treated by straigtforward considerations, and will be skipped. Lemma 3.2. Te following sequence is exact, in te sense tat te range of eac operator coincides wit te kernel of te following operator, {0} 0 V 3 = L p,0(ω, T ) (d2 ) = grad V 2 = N p 1,0(Ω, T ) were we remind tat ffl Ω : u Ω 1 Ω u. (d 1 ) =curl V 1 = P p 1(T ) Based on te above spaces, a standard conforming Finite Element metod consists in computing te unique solution (B, E ) C 0 ([0, T ]; V 1 V 2) to t B, ϕ + curl E, ϕ = 0 ϕ V 1 L 2 (Ω) (3.7) t E, ϕ c 2 B, curl ϕ = 1 ε 0 J, ϕ ϕ V 2 H 0 (curl; Ω) were J C 0 ([0, T ]; V 2 ) represents an approximation of te given current density J and, stands for te scalar product in L 2 (Ω). Note tat using te embedding curl V 2 V 1 te second equation amounts to t B + curl E = 0 (in V 1 ) (3.8) wic justifies our strong Faraday terminology. Tis space discretization is standard ans as been studied in, e.g., Ref. [20, 19, 25, 9] were te source term J is approximated wit a standard ortogonal projection on V 2, leading to define J by J, ϕ = J, ϕ for ϕ V 2. We will see tat tis approximation of te source term gives a compatible sceme. In te non-conforming case owever, we will need to use a different approximation operator, see Teorem 4.2. ffl Ω R 3.3. Projection operators and commuting diagram properties In te conforming Finite Element case te projection operators and commuting diagram properties ave been discussed and described in a series of papers by Arnold, Falk and Winter on wat tey call te Finite Element Exterior Calculus (FEEC) [1, 2, 3]. For te present paper we sall only need 96

8 Compatible Maxwell solvers wit particles, II to review te properties of π curl, te canonical projection on te curl-conforming Nédélec space. In 2D tis projection uses edge and face based degrees of freedom, and it satisfies a commuting diagram H 1 (Ω) 2 H 0 (curl; Ω) curl L 2 (Ω) V 2 π curl curl V 1 P V 1 (3.9) involving te ortogonal projection on te discontinuous space V 1, see Equation (3.12) below. As a matter of fact, te canonical projection on te Nédélec space can be derived from te standard Raviart-Tomas interpolation (recalled in our companion article [10]) by a rotation of π/2. Specifically, te degrees of freedom for te finite element space V 2 = N p 1,0(Ω, T ) read (see, e.g., [21] or [15]) { M 2 (T, u) := { M 2 (e, u) := { T u q : q P p 2(T ) 2 } for every triangle T T, e (n e u) q : q P p 1 (e)} for every edge e E \ E B (3.10) were E B denotes te set of boundary edges. Te Nédélec finite element interpolation is ten defined by te relations π curl : H 1 (Ω) 2 V 2 := N p 1,0 (Ω, T ) M 2 (T, π curl u u) = {0}, T T and M 2 (e, π curl u u) = {0}, e E \ E B. (3.11) Again, tis interpolation satisfies a commuting diagram property wic is easily verified using integration by parts, curl π curl u = P V 1 curl u, u H 1 (Ω) 2 H 0 (curl; Ω) (3.12) were P V 1 denotes te L 2 projection on te discontinuous space V 1 = P p 1(T ), and a classical error estimate π curl u u c m u m, 1 m p, u H 0 (curl; Ω). (3.13) Remark 3.3. As is well known, in 2D te Nédélec finite element space N p 1 can be obtained by rotating te Raviart-Tomas space RT p 1 by an angle of π/2: we ave N p 1 (T ) = R RT p 1 (T ) were R : u ( u y, u x ). Up to te boundary conditions (for wic te degrees of freedom need to be added or substracted from te respective bases), it is ten easily verified tat π curl = Rπ divr 1, so tat te properties listed ere for te Nédélec interpolation can be derived from tose of te Raviart-Tomas interpolation recalled in [10], using te identity curl = div R 1. Wen designing non-conforming approximations based on broken spaces in Section 4.1 it will be convenient to follow [13] as we ave done in [10] for te non-conforming strong Ampère sceme, and use a basis for V 2 tat is dual to te above degrees of freedom. Its construction goes as follows. Given scalar-valued bases q e,i, i = 1,... p for te edge polynomials P p 1 (e), e E \ E B, and vector-valued bases q T,i, i = 1,..., p(p 1) for te volume polynomials P p 2 (T ) 2, T T, we span te moment spaces listed in (3.10) wit te degrees of freedom { σ 2 T,i (u) = σe,i 2 (u) = T u q T,i, T T, i = 1,..., p(p 1) e (n e u)q e,i, e E \ E B, i = 1,..., p. (3.14) It follows from te unisolvence of (3.10) tat N p 1 (T ) admits a unique basis ϕ 2,T wit indices in { Λ Λ 2 (T ) := Λ 2 vol(t ) Λ 2 2 edge(t ) were vol (T ) := {(T, i) : i = 1,..., p(p 1)} Λ 2 edge (T ) := {(e, i) : e E(T ), i = 1,..., p} 97

9 M. Campos Pinto & E. Sonnendrücker tat is dual to te associated degrees of freedom, in te sense tat we ave σγ(ϕ 2 2,T ) = δ γ, for γ, Λ 2 (T ). To form a basis of te global space V 2 we ten gater all te indices, except for tose attaced to boundary edges, into Λ 2 := ( T T Λ 2 (T ) ) \ {(e, i) : e E B, i = 1,... p}. Te curl-conformity of (3.10) ten guarantees tat if we set ϕ 2,T := 0 for Λ 2 \ Λ2 (T ) and if we extend ϕ 2,T by 0 outside T for Λ 2 (T ), ten te piecewise polynomials ϕ 2 := 1 T ϕ 2,T = ϕ 2,T (3.15) T T T T are in H 0 (curl; Ω) and tey form a basis for te space V 2 tat is dual to te associated degrees of freedom in te sense tat σγ(ϕ 2 2 ) = δ γ, for γ, Λ 2. Moreover, if te polynomials q e,i and q T,i involved in (3.14) are defined as suitable affine maps of polynomial bases defined on a reference element, te resulting local basis functions ϕ 2,T will also correspond to affine maps of te associated reference basis. As a result, if te mes T is sape regular in te standard sense of, e.g., [15, Def. I-A.2], it is possible to ask for normalized local basis functions satisfying, e.g., ϕ 2,T 1 for T T, Λ 2 (T ). (3.16) 3.4. Gauss-compatibility of te conforming FEM-Faraday sceme In compact form, te conforming sceme (3.7) reads t U A U = F wit U = (cb, E ) T and F = (0, ε 1 0 J ) T. Te composite curl operator is defined on V := V 1 V 2 by A := c ( 0 ) curl curl 0 wit curl := curl V 2 : V 2 V 1 curl := (curl ) : V 1 V 2. (3.17) Following Definition 2.1 and considering source and auxiliary approximation operators of te form ( π 1 ) Π = 0 (ˆπ 1 ) 0 π 2 and ˆΠ = 0 0 ˆπ 2 (3.18) we ten see tat tis sceme is Gauss-compatible on some product space ˆV 1 ˆV 2 if for l = 1, 2 we can find approximation operators π l and ˆπl mapping on V l, suc tat and π 2 curl u = (curl ) ˆπ 1 u, u ˆV 1 (3.19) π 1 curl u = curl ˆπ 2 u, u ˆV 2. (3.20) Note tat tese relations read π 2d1 u = d 1 ˆπ1 u and π1 (d1 ) u = (d 1 ) ˆπ 2 u wit te notations of Section 2 and 3.2. Teorem 3.4. Te conforming FEM-Faraday sceme (3.7) associated wit an ortogonal projection for te current, namely π 2 := P V 2 : L 2 (Ω) 2 V 2 = N p 1,0 (Ω, T ), (3.21) see (3.6), is Gauss-compatible on te product space ˆV 1 ˆV 2 := V 1 (H 1 (Ω) V 2 ) 98

10 Compatible Maxwell solvers wit particles, II were V 1 = H(curl; Ω) and V2 = H 0(curl; Ω), see Section 2. In particular, Equation (3.19) olds wit an L 2 -projection ˆπ 1 := P V 1 and Equation (3.20) olds wit π1 := P V 1 and ˆπ2 := πcurl te canonical (Nédélec) interpolation on V 2 defined in (3.11). Moreover, tese mappings satisfy ˆπ 1 u u m u m, 0 m p (3.22) ˆπ 2 u u m u m, 1 m p, u H 0 (curl; Ω) (3.23) π 1 u u m u m, 0 m p (3.24) π 2 u u m u m, 0 m p. (3.25) Proof. Let us first sow tat te relation (3.19) olds as claimed, tat is, P V 2 d 1 u = d 1 P V 1 u, u V 1. (3.26) Since bot sides belong to V 2 by construction, one can test tis equality against an arbitrary v V 2, wic is in V2 by conformity (3.2). Using te definition of te various operators, in particular te fact tat (d 1 ) is defined as te restriction of (d 1 ) to V 2, we compute P V 2d 1 u, v = d 1 u, v = u, (d 1 ) v = u, (d 1 ) v = P V 1u, (d 1 ) v = d 1 P V 1u, v, wic proves (3.19). On te oter and, (3.20) is noting but te commuting diagram (3.12). Finally te error estimates are standard for L 2 projections, and (3.22) is (3.13). If one is solving te Maxwell equations wit exact sources, Teorem 2.5 from [10] applies and gives te following a priori estimate. Corollary 3.5. Let (B, E) be te exact solution to te Maxwell system (2.3). Te semi-discrete solution to te FEM-Faraday sceme (3.7) coupled wit te ortogonal projection (3.21) for te current satisfies (B B )(t) + (E E )(t) B (0) ˆπ B(0) 1 + E (0) ˆπ E(0) 2 ˆ t + m( ) B(0) m + t B(s) m ds + m ( E(0) m + for 0 m p, 1 m p, and wit a constant independent of and t. 0 ˆ t 0 ) t E(s) m ds Remark 3.6. A priori estimates leading to long-time stability are known already for te strong Faraday sceme (3.7), see [20, 19, 25]. Te main benefit of our analysis is tat it readily applies to Maxwell solvers for wic te L 2 projection is not a compatible approximation operator, and nonconforming discretizations, see e.g. Section 4.2. Remark 3.7. In te case of approximate sources, one must resort to te analysis developped in Section 2.4 of [10] to be able to derive long-time stability estimates. Tis will be done in Section 5.1 by sowing tat te FEM-Faraday sceme (3.7) is naturally structure-preserving. 4. Discontinuous elements for te 2D Maxwell system wit a strong Faraday law Because of its weak formulation, discretizing in time te Ampère law from (3.7) requires to invert a mass matrix associated wit te space V 2, and due to te curl-conformity of te latter space te resulting inversion can not be performed locally. Tis can of course become a computational burden wen te meses become very fine and wen parallel algoritms come into play. For tis reason we study a non-conforming metod were te solution is approximated in a fully discontinuous space, Ṽ 2 V 2 = H 0 (curl; Ω). 99

11 M. Campos Pinto & E. Sonnendrücker It turns out from our analysis tat a natural coice corresponds to using te broken Nédélec space Ṽ 2 := N p 1 (T ) = {u L 2 (Ω) : u T N p 1 (T ), T T } (4.1) see (3.6). Standard polynomial spaces are also a possible coice, as discussed in Remark 4.7 below. Altoug te common way for designing non-conforming discretizations is to follow te discontinuous Galerkin metodology (see e.g. [14, 18]), in tis work we aim at preserving a strong Faraday equation like (3.8). For tat purpose we experiment a different pat and apply to te 2D problem te ideas of te Conga discretization proposed and studied in [11] for te 3D Maxwell system (te name standing for Conforming/Non-conforming Galerkin ). In particular, our non-conforming discretization will be derived from te conforming one (3.3). As we did for te conforming case, and following our 3D study [11], we will sow tat tis non-conforming discretization can be equipped wit a Gauss-compatible approximation operator in te sense of Definition 2.1. In Section 5.2 we will extend tis analysis by furter verifying tat it is essentially a structure-preserving discretization of Maxwell s equations in te sense of Definition 2.2, once associated wit a nonstandard discrete divergence. Interestingly, we will see tat in te 2D setting te resulting metod can be interpreted as a centered DG metod, see Section Non-conforming Conga discretization To extend te conforming metod (3.7) on te non-conforming space Ṽ 2 operator we consider a smooting P 2 : L 2 (Ω) 2 V 2 (4.2) (wic is not required to satisfy a commuting diagram) and we define te associated Conga approximation (B, E ) C 0 ([0, T ]; V 1 Ṽ 2 ) by te system t B + curl PE 2 = 0 in V 1 t E, ϕ c 2 B, curl P 2 ϕ (4.3) = 1 ε 0 J, ϕ ϕ Ṽ 2 H 0 (curl; Ω) were again J represents an appropriate approximation (in Ṽ 2 ) of te current density J. Below we will see tat Gauss-compatible and carge-conserving scemes cannot be obtained wit a straigtforward L 2 projection as in te conforming case. For te smooting projection P 2 one may tink of using te L2 projection on te conforming space V 2, but tis would ave te downside of requiring to invert a V 2 mass matrix, a global computation tat we precisely wis to avoid. Similarly as in [11, 10] we tus use an averaging procedure based on te canonical degrees of freedom for te curl-conforming space V 2, ere (3.10). To obtain a stable projection in L 2 we can recycle te elegant construction proposed in [13], were on eac mes triangle T te autors use te local basis tat is dual to te broken basis built in Section 3.3, namely te basis, Λ2 (T ), of N p 1 (T ) tat is defined by te relations ψ 2,T ψ 2,T, ϕ2,t γ = σ(ϕ 2 2,T γ T ) = δ,γ for, γ Λ 2 (T ). (4.4) A convenient projection operator P 2 : L2 (Ω) 2 V 2 P 2 u := Λ 2 T T () is ten given by u, ψ 2,T #(T ()) ϕ2 (4.5) were T () := {T T : Λ 2 (T )} denotes te cells for wic is an active index. Using (3.15) and (4.4) we verify easily tat tis is indeed a projection on V 2 = Span({ϕ2 : Λ2 }). More 100

12 Compatible Maxwell solvers wit particles, II precisely, on te broken Nédélec space it amounts to averaging te broken version of te degrees of freedom (3.14): u N p 1 (T ) = { σ 2 T,i (P 2 u) = σ2 T,i (u T ) = σ 2 T,i (u), T T, i = 1,..., p(p 1) σ 2 e,i (P2 u) = 1 2 (σ2 e,i (u T (e)) + σ 2 e,i (u T + (e))), e E \ E B, i = 1,..., p (4.6) were we remind tat T ± (e) are te two triangles tat sare te interior edge e. Indeed, decomposing u = T T, Λ 2 (T ) ct ϕ2,t, we infer from (4.4) tat ct = σ2 γ(u T ). Te duality (3.14) gives ten σγ(p 2 u) 2 = σγ 2 c T #(T ()) ϕ2 T T = (γ) σ2 γ(u T ), for γ Λ 2 #(T (γ)), (4.7) Λ 2 T T () ence (4.6). Now, just as te basis functions ϕ 2,T can be obtained as affine maps of reference basis functions wit an L 2 normalization (3.16), it is possible to design te dual basis functions wit te same property, ψ 2,T 1 for T T, Λ 2 (T ). (4.8) In particular, P 2 is locally bounded in L2 and since it is a projection on te conforming space V 2 it satisfies an error estimate similar to (3.13). In te analysis of te structure-preserving Conga sceme, an important tool is te adjoint operator (P 2) wic is bounded on L 2 like P 2, and as te form (P 2 ) u = Λ 2 T T () u, ϕ 2 #(T ()) ψ2,t. (4.9) Lemma 4.1. Te operator (P 2 ) maps on te non-conforming space N p 1 (T ), it is locally bounded in L 2 and it preserves te piecewise polynomials of P p 2 (T ) 2. In particular, if te mes T is sape regular we ave (I (P 2 ) )u C m u m, 0 m p 1 (4.10) wit a constant independent of. Proof. Te fact tat (P 2) maps on N p 1 (T ) is obvious as every ψ 2,T is in tis space, and te local L 2 bound is easily derived using te localized supports of te basis functions ϕ 2 and te normalization of te primal ϕ 2,T s and te dual ψ2,t s, see (3.16) and (4.8). To sow tat (P2 ) preserves te piecewise polynomials of P p 2 (T ) 2 we observe tat due to te form of te volume-based degrees of freedom in (3.10), te functions ψ 2,T, = (T, i) Λ2 vol (T ), coincide wit te polynomials q T,i in (3.14), ence tey form a basis of P p 2 (T ) 2. Since te associated ϕ 2 vanis outside T, tey satisfy ϕ2 = ϕ2,t and it is easily seen tat (P 2) ψ 2,T = ψ 2,T for all Λ 2 vol (T ) (note tat every suc is in Λ2 ). Estimate (4.10) is ten a straigtforward consequence of tese properties and te Bramble-Hilbert Lemma Gauss-compatibility of te non-conforming Maxwell solver We now establis tat te above sceme can be made Gauss-compatible and give a priori error estimates leading to long-time stability. Again we denote U = (cb, E ) T and F = (0, ε 1 0 J ) T. In compact form, te non-conforming Conga-Faraday sceme (4.3) reads t U A U = F wit a 101

13 M. Campos Pinto & E. Sonnendrücker composite curl operator tat takes a form similar to (3.17) but also involves te smooting projection P 2. Specifically, it is defined on V := V 1 Ṽ 2 by ( ) 0 curl := curl P curl Ṽ 2 2 : Ṽ 2 V 1 A := c wit (4.11) curl 0 curl := (curl ) : V 1 Ṽ 2. According to Definition 2.1 and using approximation projection operators of te form (3.18), we ten see tat tis sceme is Gauss-compatible on some product space ˆV 1 ˆV 2 if for l = 1, 2 we can find approximation operators π l and ˆπl mapping on V l, suc tat π 2 curl u = (curl ) ˆπ 1 u, u ˆV 1 (4.12) and π 1 curl u = curl Pˆπ 2 u, 2 u ˆV 2. (4.13) Te following compatibility result is ten easy to verify. Teorem 4.2. Te Conga-Faraday sceme (4.3) associated wit te corrected projection operator π 2 := (P 2 ) : L 2 (Ω) 2 Ṽ 2 for te current, see (4.9), is Gauss-compatible on te product space ˆV 1 ˆV 2 := H(curl; Ω) ( H 1 (Ω) 2 H 0 (curl; Ω) ). In particular, Equation (4.12) olds wit te L 2 projection ˆπ 1 := P V 1 and Equation (4.13) olds wit π 1 := P V 1 and ˆπ2 := πcurl te Nédélec interpolation defined in (3.11). Moreover, tese mappings satisfy ˆπ 1 u u m u m, 0 m p (4.14) ˆπ 2 u u m u m, 1 m p, u H 0 (curl; Ω) (4.15) π 1 u u m u m, 0 m p (4.16) π 2 u u m u m, 0 m p 1. (4.17) Proof. Since bot sides of Equation (4.12) are in Ṽ 2, we can test it against an arbitrary v Ṽ 2. We tus compute for u ˆV 1 = H(curl; Ω) π 2 curl u, v = (P 2 ) curl u, v = curl u, P 2 v = u, curl P 2 v = P V 1 u, curl v = curl P V 1 u, v were we ave used te equality curl = curl P 2 = (curl ) : Ṽ 2 V 1, and tis proves (4.12) wit ˆπ 1 = P V 1. As for Equation (4.13), it simply follows from te commuting diagram (3.12) and te fact tat we ave curl π curl = curl P 2πcurl = curl π curl since P 2 is a projection on V 2. Estimates (4.14) and (4.16) are ten standard for te L 2 projection on V 1 and (4.15), (4.17) are (3.13) and (4.10), respectively. If one is solving te Maxwell equations wit exact sources, Teorem 2.5 from [10] applies and gives te following a priori estimate. Corollary 4.3. Let (B, E) be te exact solution to te Maxwell system (2.3). Te semi-discrete solution to te non-conforming Conga-Faraday sceme (4.3) coupled wit te corrected projection (4.9) for te current satisfies (B B )(t) + (E E )(t) B (0) ˆπ B(0) 1 + E (0) ˆπ E(0) 2 ˆ t + m( ) B(0) m + t B(s) m ds + m ( E(0) m + 0 ˆ t 0 ) ( t E(s) m ) ds 102

14 Compatible Maxwell solvers wit particles, II for 0 m p, 1 m p 1, and wit a constant independent of and t. Remark 4.4. In te case of approximate sources, one must resort to te analysis developped in Section 2.4 of [10] to be able to derive long-time stability estimates. Tis will be done in Section 5.2 by sowing tat te Conga-Faraday sceme (4.3) can be equipped wit a non-standard divergence tat makes it structure-preserving Reformulation as a standard discontinuous Galerkin sceme Because our smooting projection P 2 is defined as an averaged interpolation on te Nédélec elements V 2, it is possible to reformulate te Conga-Faraday sceme (4.3) as a standard DG sceme. To verify tis claim we remind tat a centered-flux DG approximation (see, e.g., [14]) based on te discontinuous spaces V 1 and Ṽ 2 defines (B, E ) C 0 ([0, T ]; V 1 Ṽ 2 ) as te solution to t B, ϕ + E, curl ϕ T {E }, [[ϕ] e = 0, ϕ V 1 T T e E \E B t E, ϕ c 2 B, curl ϕ T + c 2 {B }, [[ϕ] e = 1 (4.18) J, ϕ, ϕ ε Ṽ 2. T T e E 0 Here we ave used standard notations for tangential jumps and averages (see, e.g., [6]): for interior edges (sared by two cells T ± = T ± (e), and writing n ± e = n T ± e for simplicity ) we denote [[u] e := (n e u T + n + e u T +) e and {u } e := 1 2 (u T + u T +)) e for e E \ E B (4.19) and for boundary edges (in te boundary of a single cell T = T (e)), [[u] e := (n e u T ) e and {u } e := (u T ) e for e E B. (4.20) For a scalar-valued u te definitions are formally te same, keeping in mind tat wit te 2D convention te product n u is te vector (n y u, n x u) T. To write (4.18) in an operator form we ten let curl DG : Ṽ 2 V 1 and curl DG : V 1 Ṽ 2 (4.21) be defined by te relations curl DG u, v := u, curl v T {u }, [[v ] e T T e E \E B curl DG v, u := v, curl u T {v }, [[u] e T T e E for v V 1, u Ṽ 2. (4.22) Hence writing again U = (cb, E ) T and F = (0, ε 1 0 J ) T, te DG formulation (4.18) reads ( ) 0 curl DG t U A U = F wit A := c curl DG. 0 Te following results establises tat tis approximation is equivalent wit te Conga metod (4.3). Teorem 4.5. Te DG curl operators defined above satisfy curl DG = curl P 2 on Ṽ 2, and curl DG = (curl DG ). In particular, te Conga-Faraday sceme (4.3) is equivalent wit te centered-flux DG sceme (4.18). 103

15 M. Campos Pinto & E. Sonnendrücker Remark 4.6. Tis result is specific to te 2D setting, for several reasons. First, te different degrees involved in te edge and face degrees of freedom prevent similar computations to be carried out in 3D. In particular, it is not true tat in 3D te centered DG sceme involves a strong Faraday law, as tis would imply tat te divergence of te magnetic field remains constant in time, a property known to be false in general [24]. Anoter indication is offered by considering te spectral properties of te respective metods: wereas te 3D Conga metod as been proved to be spectrally correct in [11], numerical and teoretical evidences [16] sow tat tis is not te case for te centered DG sceme. Proof. To prove te first equality we compute for v V 1 and u Ṽ 2, curl Pu, 2 v = ( P 2 u, curl v T + n Pu, 2 ) v T T T = T T = ( u, curl v T + n {u }, v T \ Ω ) T T u, curl v T e E \E B {u }, [[v ] e = curl DG u, v. Here te second equality follows from te property (4.6) of P 2 and te form (3.14) of te Nédélec degrees of freedom, togeter wit te fact tat v T P p 1 (T ) wic gives curl v T P p 2 (T ) 2 and v e P p 1 (e) for e E(T ). Te desired equality curl DG = curl P 2 ten follows from te fact tat curl P 2 maps on V 1. Starting next from te second line of (4.22) and integrating by parts we compute curl DG v, u = v, curl u T {v }, [[u] e T T e E = ( ) curl v, u T + v, n u T {v }, [[u] e T T e E = ) u, curl v T + ( v, n u e + v +, n + u + e {v }, [[u] e T T e E \E B = u, curl v T {u }, [[v ] e = curl DG u, v T T e E \E B and te desired equality follows from te definition of te adjoint. Note tat in te tird equality we ave used te fact tat {v }, [[u] e = v, n u e on every boundary edge e E B. Remark 4.7. Discontinuous Galerkin scemes are more commonly used wit standard polynomials spaces, suc as Ṽ 2 := P p 1 (T ) 2 = {u L 2 (Ω) : u T P p 1 (T ) 2, T T }. (4.23) To apply our analysis to tat case te most natural pat consists in replacing te conforming sequence (3.3) by te following one V 3 = L p,0 (Ω, T ) (d2 ) = grad V 2 = P p 1 (T ) 2 H 0 (curl; Ω) (d1 ) = curl V 1 = P p 2 (T ) wic is also exact. Here te space V 2 corresponds to te Nédélec elements of second type. Accordingly one replaces te degrees of freedom (3.10) by {ˆ ( } M 2 (T, u) := u q : q P p 3 (T ) 2 x + P T y) p 3 (T ) for every triangle T T, {ˆ } (4.24) M 2 (e, u) := (n e u) q : q P p 1 (e) for every edge e E \ E B e 104

16 Compatible Maxwell solvers wit particles, II see, e.g., [22, 4] and a new smooting projection P 2 based on tese degrees of freedom can be designed following te same steps as before. One can verify tat te resulting Conga sceme is equivalent wit te associated DG metod. Namely, Teorem 4.5 olds wit Ṽ 2 = P p 1(T ) 2 and V 1 = P p 2(T ). In tis article we ave worked wit te first Nédélec space because it as better convergence properties tan te second one (see e.g. [5]), but we believe tat a proper teoretical and numerical study of te oter option sould also be performed. 5. Application to te coupled Vlasov-Maxwell problem In tis section we apply te new stability analysis proposed in Section 2.1 and 2.4 of [10] for approximate sources: in Section 5.1 we begin by verifying tat te conforming Finite Element discretization studied in Section 3 is naturally structure-preserving in te sense of Definition 2.2, and in Section 5.2 we sow tat our new non-conforming Conga discretization of Section 4 is also structure-preserving, once associated wit a nonstandard discrete divergence. Assuming next a discrete particle representation of te approximate current density, for eac Maxwell solver we provide a carge-conserving current deposition metod in te sense of Definition 2.3. To specify te problem we consider te case were te Maxwell system is coupled wit a Vlasov equation suc as (1.2) involving a species of carged particles wit pase space distribution function f = f(t, x, v). Te carge and current densities are ten given by te first moments of f, ˆ ˆ ρ(t, x) := q f(t, x, v) dv and J(t, x) := q vf(t, x, v) dv. (5.1) 5.1. Structure-preserving discretization wit conforming Finite Elements Te structure-preserving properties of te conforming Maxwell sceme (3.7) essentially follow from grad curl ffl te fact tat V 3 V 2 V 1 Ω R is an exact sequence, as recalled in Lemma 3.2. Te Poincaré estimates (2.9) are also standard to verify. Te first one reads u c P grad u, u V 3 (5.2) and is a standard Poincaré inequality, given te omogeneous boundary condition. Te second one u c P curl u, u V 2 (ker curl) (5.3) can be derived, e.g. from te similar stability estimate [23, T. 4] recalled in [10, Eq. (5.3)] for te Raviart-Tomas elements, using te standard rotation argument of Remark 3.3. Finally te tird one involves te integral operator ffl Ω and trivially olds on V 1 (ker ffl Ω ) R. Hence te Lemma. Lemma 5.1. Te conforming sceme (3.7) associated wit te discrete Gauss laws (2.7) defined by (ι ) := : V 1 R Ω grad := grad V 3 : V 3 V 2 (5.4) div := (grad ) : V 2 V 3 see (3.3), is structure preserving in te sense of Definition 2.2. Remark 5.2. Wit te operators (5.4), te discrete Gauss laws (2.7) read E (t), grad φ = 1 ρ (t), φ for φ V 3 ε 0 B (t) = Ω B. 0 Ω (5.5) 105

17 M. Campos Pinto & E. Sonnendrücker 5.2. Structure-preserving discretization wit te discontinuous Conga metod To study te structure-preserving properties of te Conga metod, and identify a proper discrete divergence, we first caracterize te kernel and te image of te non-conforming curl operator, following te metod introduced in [7]. Lemma 5.3. Te non-conforming curl operator (4.11), curl = curl P 2 Ṽ 2 : Ṽ 2 V 1, satisfies ker(curl ) = grad V 3 (I P 2 )Ṽ 2 and Im(curl ) = V 1 R. Proof. Starting wit te first identity, te inclusion is verified by applying curl P 2 and te fact tat grad V 3 is a subset of V 2 were P2 = I. To verify te inclusion we take u Ṽ 2 ker(curl P2 ). Ten P 2u is in V 2 ker curl wic coincides wit grad V 3 tanks to Lemma 3.2. Hence we ave u = Pu 2 + (I P)u 2 grad V 3 (I P)Ṽ 2 2, and we easily verify tat tis is an ortogonal sum. Te second identity follows from Lemma 3.2 and te fact tat Ṽ 2 contains V 2, ence P2 Ṽ 2 = V 2. We are ten in position to establis tat te Conga-Faraday sceme is structure preserving wen associated wit te proper discrete operators for te Gauss laws. Lemma 5.4. Te non-conforming Conga sceme (4.3) associated wit te discrete Gauss laws (2.7) defined by (ι ) := : V 1 R Ω grad : (V 3 Ṽ 2 ) (φ, ũ) grad φ + (I P)ũ 2 Ṽ 2 (5.6) div := (grad ) : Ṽ 2 (V 3 Ṽ 2 ) see (3.3), is structure preserving in te sense of Definition 2.2. Remark 5.5. Wit te proposed operators (5.6), te discrete Gauss laws (2.7) read E (t), (grad φ + (I P)ũ) 2 = 1 ρ (t), φ for (φ, ũ) V 3 ε Ṽ 2 0 B (t) = Ω B. 0 Ω Proof. Here te exact sequence property (2.8) reads V 3 Ṽ 2 grad Ṽ 2 curl =curl P 2 V 1 Ω R (5.8) and it follows from Lemma 5.3. To prove te stability estimates in (2.9) we follow te proof of Teorem 4.1 from [8]. We begin by observing tat since grad V 3 (I P2 )Ṽ 2 is a direct sum, one as ker grad = (V 3 ker grad) (Ṽ 2 ker(i P)) 2 = {0} V 2. Considering ten φ V 3, ũ Ṽ 2 (V 2) and using (5.2) we compute (φ, ũ) 2 c P grad φ 2 + ũ 2 grad φ P ũ P ũ ũ 2 ffl (5.7) grad φ P 2 ũ 2 + ũ 2 = grad φ + (I P 2 )ũ 2 were te last equality uses tat grad φ Pũ 2 is in V 2 and ence is ortogonal to ũ. Tis is te first estimate in (2.9). For te second estimate, we use again te identity ker curl = grad V 3 (I P2 )Ṽ 2 and consider now ũ Ṽ 2 (ker curl ) = Ṽ 2 (grad V 3 ) ((I P)Ṽ 2 2 ) 106

18 Compatible Maxwell solvers wit particles, II and let u V 2 (ker curl) be defined by curl u = curl P ũ. 2 Tis implies tat te difference u P2 ũ is in V 2 ker curl = grad V 3, ence it is ortogonal to ũ. Because te latter is also ortogonal to (I P 2 )ũ, we find tat it is ortogonal to ũ u. Using tis and te conforming Poincaré estimate (5.3) for u we compute ũ ( ũ 2 + u ũ 2 ) 1 2 = u cp curl u = c P curl P ũ 2 wic proves te non-conforming Poincaré estimate. Finally te tird estimate is te same as in te conforming case, and te proof is complete Carge-conserving coupling wit smoot particles In te particle metod te pase-space distribution function f solution to (1.2) is approaced by a sum of N (macro) particles wit positions x κ (t) and velocities v κ (t) = x κ(t), κ = 1,... N, tat are pused forward along te integral curves of te semi-discrete force field computed by te Maxwell sceme, using some given ODE solver. Te approximated density is ten N f N (t, x, v) = q κ ζ ε (x x κ (t))ζ ε (v v κ (t)) (5.9) κ=1 were q κ is te numerical carge associated wit te κ-t (macro) particle and ζ ε is a sape function supported in te Ball B(0, ε) of center 0 and radius ε 0, wic can eiter be a smoot approximation of te Dirac measure if ε > 0 (typically a spline wit unit mass, see e.g. [18]) or te Dirac measure itself if ε = 0. Te corresponding approximations for te carge and current densities read ten N N ρ N (t, x) := q κ ζ ε (x x κ (t)) and J N (t, x) := q κ v κ (t)ζ ε (x x κ (t)). (5.10) κ=1 We observe tat since v κ (t) = x κ(t), tese particle densities satisfy an exact continuity equation, N div J N = q κ div ( v κ ζ ε ( x κ ) ) N N = q κ v κ grad ζ ε ( x κ ) = q κ t ζ ε ( x κ ) = t ρ N. (5.11) κ=1 κ=1 κ=1 Note tat in te case were ε = 0 tese equalities old in a weak sense, as we ave N N N J N, grad φ = q κ v κ (t) grad φ(x κ (t)) = q κ t φ(x κ (t)) = q κ t δ xκ(t), φ = t ρ N, φ (5.12) κ=1 κ=1 κ=1 for φ C 2 (Ω) In order to make bot te conforming and te non-conforming scemes carge conserving in te sense of Definition 2.3 we must ten find proper approximations J for te particle current J N. Te following result sows tat for tis task we can use te ortogonal projection in te conforming case and te corrected projection in te non-conforming case, just as for te compatibility results stated in Sections 3.4 and 4.2. Teorem 5.6. Let ε > 0. Te respective conforming (FEM) and non-conforming (Conga) scemes (3.7) and (4.3), associated wit te discrete Gauss laws (2.7) defined by te discrete divergence operators (5.4) and (5.6) respectively, are carge conserving in te sense of Definition 2.3 wen te discrete sources are defined from te particle carge and current densities (5.10) by in te conforming case and ρ (t) := P V 3 ρ N (t) V 3 and J (t) := P V 2 J N (t) V 2 (5.13) ρ (t) := P V 3 ρ N (t) V 3 and J (t) := (P 2 ) J N (t) Ṽ 2 (5.14) κ=1 107

19 M. Campos Pinto & E. Sonnendrücker in te non-conforming case. Here P V 3 and P V 2 are te L 2 (ortogonal) projections on te continuous and Nédélec spaces respectively, and (P 2 ) is te discrete adjoint of te smooting projector, see (4.9). Remark 5.7. As will be explained in Section 5.5, te compatible current deposition proposed ere for te Conga metod only involves local computations. Tis is a significant difference wit te conforming case were te L 2 projection requires to invert a mass matrix in te curl-conforming space V 2, wic is a global operation. Proof. Since we already know tat (3.7) and (4.3) are structure preserving wen associated wit te respective operators (5.4) and (5.6), it suffices to verify tat te resulting discrete continuity equation (2.15) indeed olds in bot cases. In te conforming case were div is defined by its adjoint (div ) = grad : V 3 V 2 te discrete continuity equation reads J, grad φ = t ρ, φ for φ V 3 (5.15) and is easily verified for ε > 0 by computing as in (5.12) wit φ V 3, since ten bot J N(t) and ρ N (t) are in L 2 (Ω). In te non-conforming case te discrete operator div is defined by its adjoint (div ) = grad : V 3 Ṽ 2 Ṽ 2 as in (5.6), and te discrete continuity equation reads J, grad (φ, ũ) = J, grad φ + (I P)ũ 2 = t ρ, φ for (φ, ũ) V 3 Ṽ 2. (5.16) In particular, plugging J = (P 2) J N in te above formula and using te fact tat grad V 3 V 2 yields J, grad (φ, ũ) = J N, P 2 grad φ = J N, grad φ, so tat te desired equality follows as in te conforming case Carge-conserving coupling wit point particles To extend our results to te case of point particles (ε = 0) it is convenient to consider a fully discrete version of te proposed Maxwell solvers. For simplicity we assume an explicit leap-frog time sceme. For te conforming (FEM) metod (3.7) te approximate fields (B n+1/2, E n) V 1 V 2 are ten given by B n+ 1 2 B n t curl E n = 0 (in V 1 ) (5.17) E n+1 E, n ϕ c 2 t B n+ 1 2, curl ϕ = t ε 0 J n+ 1 2, ϕ ϕ V 2 and for te non-conforming (Conga) metod (4.3) te discrete fields (B n+1/2, E n ) V 1 Ṽ 2 are updated wit B n+ 1 2 B n t curl P 2 E n = 0 (in V 1 ) E n+1 E, n ϕ c 2 t B n+ 1 2, curl P 2 ϕ = t ε 0 J n+ 1 2, ϕ ϕ Ṽ 2. (5.18) In bot cases we tus need to define J n+1/2 from te current density J N carried by te moving particles. Following our stability analysis we would like tat te resulting solutions satisfy te proper discrete Gauss laws wic involve te structure-preserving divergence operators identified in tis work, namely (5.5) and (5.7) respectively. In [9] tis construction was described for te conforming FEM metod, using a time averaging and an extension of te L 2 projection (5.13) for current densities carried by Dirac particles. Specifically, it was sown tat te quantities J n+ 1 2, ϕ = ˆ tn+1 t n J N (τ) dτ t, ϕ ˆ N tn+1 = q κ v κ (τ) ϕ(x κ (τ)) dτ κ=1 t n t, ϕ V 2, (5.19) 108

20 Compatible Maxwell solvers wit particles, II are well defined for point particles, and allow to define a current J n+1/2 V 2 wic satisfies a time-discrete version of te proper continuity equation (5.15) in te conforming case, i.e., J n+ 1 2, grad φ = 1 1 ε 0 t (ρn+1 ρ n ), φ φ V 3. (5.20) Here ρ n V 3 is defined by te relations N ρ n, φ = ρ N (t n ), φ = q κ φ(x κ (t n )), φ V 3. (5.21) κ=1 Essentially, te reason wy (5.19) makes sense (and is stable wit respect to te particle trajectories) is tat te test functions ϕ in te curl-conforming space V 2 ave teir tangential components tat are continuous across interelement edges. In particular, we observe tat in te last integral te discontinuities of ϕ may only pose a problem wen te particle trajectory x κ runs along an edge during some non-zero time interval. But in tis case te particle velocity is tangent to te edge and te function v κ (τ) ϕ(x κ (τ)) is well defined, i.e., stable wit respect to te trajectory. For te same reason, it is possible to extend te corrected projection (P 2) in (5.14) wen te current is carried by point particles. Specifically, for te fully discrete Conga sceme (5.18) we define a carge-conserving current J n+1/2 Ṽ 2 by te relations ˆ J n+ 1 tn+1 2, ϕ = J N (τ) dτ ˆ N tn+1 t, P2 ϕ = q κ v κ (τ) (P 2 ϕ)(x κ(τ)) dτ t, ϕ Ṽ 2 (5.22) t n κ=1 since ten P 2 ϕ is curl-conforming altoug ϕ was fully discontinuous. Notice tat an ortogonal projection on te fully discontinuous space Ṽ 2 would involve products of te form v κ (τ) ϕ(x κ (τ)) dτ wic are not well-defined for particles running along te edges of te mes. Arguing next as in te proof of Teorem 5.6, one easily verifies tat te resulting sources satisfy indeed te proper continuity equation (5.16), namely wit ρ n V 3 t n J n+ 1 2, grad φ + (I P)ũ 2 = 1 1 ε 0 t (ρn+1 ρ n ), φ (φ, ũ) V 3 Ṽ 2 (5.23) defined again by (5.21). Remark 5.8. Wen te particle trajectories are piecewise polynomials as is usually te case, it is possible to compute exactly te time integrals in (5.19) and (5.22) using Gauss quadratures tat involve a few points witin eac cell travelled by te particles. We refer to [9] for te detailed algoritms Compatible current deposition seen as a correction metod Before turning to te numerical experiments, let us make two simple but important observations. First, given a cell-wise basis for Ṽ 2 tat we may denote as ϕ T, wit T T and Λ 2 (T ), we find tat te coefficients J T, of te compatible current J := (P 2) J N are determined by te relations J T, ϕ T,, ϕ T,γ = J, ϕ T,γ = J N, Pϕ 2 T,γ for T T, γ Λ 2 (T ). (5.24) Λ 2 (T ) Tus, te proposed deposition metod involves (i) computing te products of te smoot particle current against te averaged basis functions, and (ii) inverting te local mass matrices associated wit te discontinuous basis, namely ( ) M T = ϕ T,, ϕ T,γ.,γ Λ 2 (T ) 109

21 M. Campos Pinto & E. Sonnendrücker In particular we see tat tese two steps can be performed locally, unlike in te conforming case (5.13) were te inversion of a mass matrix of V 2 is always a global operation over te mes. Second, we observe tat step (i) above is easily obtained from te standard DG current coefficients wen tey are available in an existing code. Here we refer to te current defined by a standard L 2 projection on te fully discontinuous space, namely J nc := PṼ 2 J N were te exponent nc stands for non-compatible as tis projection does not satisfy te discrete continuity equation tat we ave identified in te non-conforming case. Te coefficients JT, nc of te latter in te cell-wise basis are ten determined by te relations JT, ϕ nc T,, ϕ T,γ = J nc, ϕ T,γ = J N, ϕ T,γ. (5.25) Λ 2 (T ) Now, since te averaging projection P 2 maps on V 2 wic is a subspace of Ṽ 2 by construction, it can be represented by a matrix P satisfying Pϕ 2 T, = T T,γ Λ 2 (T ) P (T,),(T,γ)ϕ T,γ so tat te compatible moments m T, (J N ) := J N, P 2 ϕ T,γ are easily derived from teir standard (non-compatible) counterparts m nc T, (J N) := J N, ϕ T,γ as m T, (J N ) = T,γ P (T,),(T,γ)m nc T,γ(J N ). (5.26) If te standard moments m nc of te particle current are available, te above formula suffices to perform step (i) and deposit te current in a compatible way. We may also verify tat te coefficients of te compatible current can be expressed as a local correction of te non-compatible ones, witout referring to te moments temselves: using global notations for arrays and matrices defined over te wole mes, Equation (5.25) reads MJ nc = m nc (J N ), wereas (5.24) and (5.26) give MJ = m(j N ) = Pm nc (J N ). Terefore we ave J = M 1 PMJ nc. Here te matrix M as a block diagonal structure corresponding to te mes cells, wereas P as entries corresponding to couples of adjacent cells, in addition to tose on te diagonal. In particular its application is indeed a local operation, wit coefficients easily given by (4.5): using local bases for te broken Nédélec spaces tat derive from te curl-conforming ones as described in Section 3.3, te non-zero entries of P are simply 1 for te volume-based degrees of freedom and 1 2 for te adjacent edge-based degrees of freedom. 6. Numerical results In tis section we illustrate te proposed FEM and Conga metods on te two test cases already used in our companion article [10]. For te time discretization we use te explicit leap-frog sceme described in Section 5.4. Remark 6.1. Wen studying te Conga metod we ave observed tat te smooted field P 2En was more accurate tan te discontinuous field E n itself: for te studied cases, it ad smaller errors and iger convergence rates. Terefore we ave decided to use te smooting projection P 2 as a systematic post-processing filter. Since tis is a local operation on te discrete fields, its effect on te overall computational time is not significant. 110

22 Compatible Maxwell solvers wit particles, II 6.1. A Pure Maxwell problem: te 2D Issautier test case To assess te basic convergence and stability properties of te proposed scemes we use te analytical current source proposed in [17, 12] to study te carge conservation properties of a penalized finite volume sceme. Te problem is posed in a metallic cavity Ω = [0, 1] 2 wit articifial permittivity ε 0 and ligt speed c equal to one, and te current density is given as J(t, x, y) = (cos(t) 1) ( π cos(πx) + π 2 ) ( ) x sin(πy) x sin(πy) π cos(πy) + π 2 cos(t). (6.1) y sin(πx) y sin(πx) We consider initial fields E 0 = 0 and B 0 = 0, so tat te exact solution is ( ) x sin(πy) E(t, x, y) = sin(t) y sin(πx) B(t, x, y) = (cos(t) 1) ( πy cos(πx) πx cos(πy) ). (6.2) We note tat te associated carge density reads ten ρ(t, x, y) = sin(t) ( sin(πx) + sin(πy) ). In Figure 6.1 we first assess te convergence properties of te two proposed metods by plotting te relative L 2 errors e := max ( E E / E, B B / B ) at time t = 0.2π. In te left plot we sow te results obtained wit te conforming FEM (3.7) using different degrees and in te rigt plot we sow te errors corresponding to te non-conforming Conga metod (4.3). Te convergence rates of bot te FEM and Conga solutions is in agreement wit Corollary 3.5 and 4.3. More precisely, we observe tat te Conga solutions of degree p + 1 (smooted as described in Remark 6.1) converge wit a similar rate tan te conforming solutions of degree p, close to p. We also note tat te former as iger accuracy. Time wise, we ave observed tat wit our straigtforward implementation te Conga simulations were more efficient tan te FEM ones wen te meses became finer, wic is not surprising since te former is purely local and does not require any global matrix inversion. Specifically, our simulations ave sown tat for p > 1, te computational time of te FEM metod wit degree p becomes iger tan tat of te Conga metod wit degree p + 1, as soon as te mes as more tan about 6000 triangles (wic corresponds to 0.06 for te meses used ere). In order to assess te long-time properties of te Conga sceme (4.3) associated wit a corrected projection for te current density, J = (P 2 ) J Ṽ 2 (6.3) as supported by our analysis, we ave plotted in te left panel of Figure 6.2 te L 2 norm of an electric field computed wit tat metod. On te rigt panel we ave sown te norm of te electric field obtained wit te same sceme (4.3) but wit a discrete current density computed by an ortogonal projection on te broken Nédélec space Ṽ 2, namely J = PṼ 2 J Ṽ 2. (6.4) In bot cases te broken Nédélec space (4.1) is defined wit p = 2, using a mes wit about 250 triangles ( 0.3). In te latter case a rapid (linear in time) deterioration of te solution is visible, but wit te Gausscompatible sceme te solution is stable, as predicted by Corollaries 2.6 and 2.7 in [10], applied to te constant and time-armonic parts of te Issautier field (6.2). Note tat ere we ave only sown te curves of te electric field, as tose of te magnetic field were always on top of te reference curves (dased) computed from te exact solutions. 111

23 M. Campos Pinto & E. Sonnendrücker Figure 6.1. Convergence curves (relative errors vs. maximal triangle diameter ) for te Issautier problem wit analytical source (6.1). Results obtained wit te conforming FEM discretization are sown left, and tose obtained wit te non-conforming Conga discretization are sown rigt Figure 6.2. Evolution of te L 2 norm of te electric field for te Issautier problem. On te left plot te numerical solution is obtained by approximating te current density wit a corrected projection on te fully discontinuous space Ṽ 2, see (6.3), wereas on te rigt plot te current density is approximated using a standard L 2 projection (6.4). For comparison, te norm of te exact solution is sown in dased lines (on te left plot it is on top of te solid line) A Vlasov-Maxwell problem: an academic diode test case Turning to te coupled FEM-PIC and Conga-PIC scemes, we use again te academic diode test case employed in [10] to test particle scemes coupled wit Maxwell solvers wit a strong Ampère law. Here te domain is a square Ω = [0, 0.1m] 2 wit metallic boundary Γ M = {0, 0.1m} [0, 0.1m] and absorbing boundary Γ A =]0, 0.1m[ {0, 0.1m}. On te left boundary a beam of electrons is steadily injected and accelerated by a constant external field wic derives from te electric potential imposed on bot te catode (φ ext = 0 on te left boundary) and te anode (φ ext = 10 5 V on te rigt boundary). Due to te propagation of te beam into te domain (initially empty of carges) a self-consistent electromagnetic field develops and is added to tis constant external field, and in turn te trajectories of te electrons are no longer straigt lines. However tis modification is of small relative amplitude and te resulting solution tends towards a smoot steady state, so tat te convergence of te numerical approximations can be easily assessed. In Figure 6.3 we sow te typical profile of te solution in te steady state regime (self-consistent electric field on te left and particles on te rigt), togeter wit te mes used in te simulations. 112

24 Compatible Maxwell solvers wit particles, II To avoid using expensive numerical quadratures in space we consider a coupling wit point particles, as described in Section 5.4. Te numerical algoritms tested ere may ten be seen as an extension of tose proposed in [9] to te case of fully discontinuous elements. Figure 6.3. Academic beam test case. Te self-consistent E field (left plot) and te numerical particles accelerated towards te rigt boundary (rigt plot) sow te typical profile of te solution in te steady state regime. For te considered geometry te external field is constant E ext = ( 10 6, 0)Vm 1. To finally assess te numerical stability properties of te proposed FEM and Conga metods over long time ranges we plot in Figure 6.4 te profiles of several fields using a final time cosen so tat te particles ave travelled approximatively five diode lengts. On te left column of Figure 6.4 we plot te profiles of te electro-magnetic field computed wit te conforming FEM-PIC sceme (5.17) using a standard L 2 projection for te particle current, wic consists in defining J n+1/2 := P V 2J n+1/2 N troug products of te form (5.19). Te stability of suc a coupling is supported by Teorem 5.6, and indeed te numerical results sow a very good preservation of te smoot steady state, bot for te electric field (top and center row) and te magnetic field (bottom row). On te center and rigt columns of Figure 6.4 we ten plot te fields computed wit te non-conforming Conga-PIC sceme (5.18) using two different deposition metods for te current, similarly as wat was done (starting from an analytical expression for J) in Figure 6.2. On te center column te DG current is obtained wit a standard L 2 projection of te particle current, i.e., J n+1/2 := PṼ 2J n+1/2 N, and on te rigt column it is defined as J n+1/2 := (P 2) J n+1/2 N. Note tat in practice tis may be done eiter by computing terms of te form (5.19) and inverting te block-diagonal DG mass matrix of Ṽ 2, or by correcting locally te array of coefficients computed wit te standard metod, as described in Section 5.5. Again, te enanced stability of te former coupling is supported by Teorem 5.6. Tis is clearly confirmed by our numerical simulation. Wereas te carge-conserving Conga sceme yields results comparable to te conforming metod, te electric field resulting from te standard DG deposition sceme as erratic oscillations tat grow linearly in time and reac, in te test done ere, values about four times greater tan te maximum amplitude of te correct solution. 7. Conclusion In tis series of papers we ave provided a rigorous solution to te longstanding problem of cargeconserving coupling between general Maxwell solvers and particle metods, following te classical approac developped by plasma pysicists over te last decades. Our stability analysis extends a recent work on compatible source approximation operators for pure Maxwell solvers, and it is based on 113

25 M. Campos Pinto & E. Sonnendrücker Figure 6.4. Academic beam test-case. Snapsots of te self-consistent fields (Ex on te top row, Ey on te center row and B on te bottom row) obtained by depositing te conservative current density carried by te particles wit eiter te conforming FEMPIC sceme wit standard L2 projection for te particle current (left), te DG-PIC sceme wit standard L2 projection for te current (center) and te DG-PIC sceme wit te corrected projection (6.3) for te particle current (rigt). te notion of discrete de Ram structure. Tis abstract setting allows us to design carge-conserving deposition scemes for general conforming but also non-conforming Maxwell discretizations, tus offering an interesting alternative to divergence cleaning metods to stabilize Discontinuous Galerkin (DG) Particle-in-Cell solvers. Te framework of de Ram sequences also allows te coice of discretizing eiter te Ampère or te Faraday equation strongly, te oter being andled by duality. In tis paper we provided te 114