Technische Universität Graz

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1 Technische Universität Graz Analysis of a kinematic dynamo model with FEM BEM coupling W. Lemster, G. Lube, G. Of, O. Steinbach Berichte aus dem Institut für Numerische Mathematik Bericht 0/3

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3 Technische Universität Graz Analysis of a kinematic dynamo model with FEM BEM coupling W. Lemster, G. Lube, G. Of, O. Steinbach Berichte aus dem Institut für Numerische Mathematik Bericht 0/3

4 Technische Universität Graz Institut für Numerische Mathematik Steyrergasse 30 A 800 Graz WWW: c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors.

5 Analysis of a kinematic dynamo model with FEM BEM coupling W. Lemster, G. Lube, G. Of, O. Steinbach Institut für Numerische und Angewandte Mathematik, Universität Göttingen Lotzestrasse 6 8, Göttingen, Germany lube@math.uni-goettingen.de Institut für Numerische Mathematik, TU Graz, Steyrergasse 30, 800 Graz, Austria of@tugraz.at, o.steinbach@tugraz.at Abstract We consider a kinematic dynamo model in a bounded interior region Ω and in an insulating exterior region Ω c := R 3 \Ω. In the so called direct problem, the magnetic field B and the electric field E are unknown, and which are driven by a given incompressible flow field w. After eliminating E, a vector and a scalar potential ansatz for B in the interior and exterior domains, respectively, are applied, leading to a coupled interface problem. We apply a finite element approach in the bounded interior domain Ω, whereas a symmetric boundary element approach in the unbounded exterior domain Ω c is used. We present results on the well posedness of the continuous coupled variational formulation, prove well posedness and stability of the semi discretized and fully discretized schemes, and provide quasi optimal error estimates for the fully discretized scheme. Finally we present some first numerical results. Introduction Large scale planetary and stellar magnetic activities are driven by magnetohydrodynamic dynamo processes in the interior of planets and stars. In the convectively unstable case, they consist of large scale global circulations and small scale turbulent flows. A widely accepted theory for the generation of large scale magnetic fields through the effect of small scale turbulences in a conducting field is the mean field dynamo theory [7] which will be considered here.

6 Starting point are the full Maxwell equations E = t B, H = j + D, B = 0, D = ϱ, t where the dielectric displacement D and the magnetic field intensity H are expressed through the electric field E and the magnetic field B, respectively, by the constitutive relations D = εe, H = µ B with the magnetic permeability µ and the electric permittivity ε. If w is a given incompressible flow field, we can write an extended Ohm s law as Rf j = σ E + w B + + s B B, where σ is the electric conductivity. The nonlinear saturation term is a turbulence modelling term from the mean field dynamo theory [6, Section 6.] where the turbulent electromotive force E = ŵ ˆB is a key quantity. Here indicates an average in the dynamo domain, and ŵ and ˆB denote the fluctuating small scale velocity and magnetic fields. Following [4] we consider the α quenching approximation E αb = Rf + s B B with a model oriented function f, and positive parameters s and R. Moreover, following [6, Section.5] we neglect the Gauss law since the electric charge ϱ is very small, and we neglect the temporal change of the dielectric displacements. Finally we assume that the electric conductivity σ and the magnetic permeability µ do not depend on time but satisfy µ µ min > 0 and 0 < σ min σ σ max in a bounded interior domain Ω. Since the exterior domain Ω c := R 3 \ Ω is assumed to be insulating, we have σ = 0 and µ = µ 0 is constant in Ω c. Hence we end up with the coupled problem E = t B, E µ B = σ Rf + w B + + s B B, B = 0 in Ω,. and E = t B, µ 0 B = 0, B = 0 in Ω c,. together with the transmission conditions [n H n] Γ = 0, [B n] Γ = 0 on Γ := Ω,.3 where [ ] Γ denotes the jump of a function across Γ, and n is the exterior normal vector. Moreover, we have some initial condition B0 = B 0, B 0 = 0.

7 Let us briefly comment on the literature. The majority of stellar and planetary dynamo models employs spectral methods with spherical harmonics which are not feasible in the case of variable data. For bounded domains, the standard approach for the finite element solution of Maxwell s system is to use curl conforming elements, e.g., Nédélec elements, see [, 3]. Alternatively, a saddle point approach with a vanishing pressure like Lagrange multiplier allows to apply Lagrangian finite elements [4]. In the case of discontinuous data, an interior penalty approach together with Lagrange finite elements is adressed in [7, 8, 4]. For the full space R 3, a usual approach is a coupling of finite element FEM or finite volume FVM methods in a bounded domain with boundary element BEM methods in the remaining exterior domain. For a FVM BEM coupling we refer, e.g., to [, 4]. The idea of a symmetric FEM BEM coupling for the stationary linear Maxwell system can be found in [3] and [0] which will be used in this paper as well. An hp adaptive FEM BEM coupling is considered in [3]. A comparison of the interior penalty finite element approach and a FVM BEM coupling is considered in [9]. In the present paper we consider the coupled problem..3 where we first eliminate the electric field E and apply a vector potential ansatz for the magnetic field B in the interior domain Ω, and a scalar potential ansatz for B in the exterior domain Ω c. The coupling of both subproblems at the interface is accomplished by using both boundary integral equations of the exterior Calderón projection where we discuss two different approaches. Then we analyze the well posedness of the arising continuous coupled problem which is formed by a time dependent nonlinear problem in the interior, and a quasi stationary elliptic problem in the exterior domain. For the spatial discretization we first introduce an approximation of the exterior Dirichlet to Neumann map, and apply lowest order Nédélec elements for a finite element discretization. For simplicity we only consider an implicit Euler scheme for the time discretization, but we prove well posedness of the discrete system arising within each time step. Finally we present a priori error estimates of the fully discretized system, and we present some first numerical results. Variational formulation In this section, we specify the mathematical model and derive a variational formulation of the coupled problem..3. Later on, and following the approach in [3], this will be the basis of a symmetric coupling of finite element methods FEM and boundary element methods BEM in the discrete case. We start by using a vector potential ansatz to describe solutions of the interior problem., and a scalar potential ansatz for the exterior problem.. 3

8 . Interior problem In the interior bounded domain Ω we use a vector potential ansatz to describe the magnetic field B = A. From. we then obtain B = A = 0, t B + E = t A + E = 0, from which we further conclude E = t A φ for some scalar potential φ. Hence we may introduce the new potential from which we conclude u := A + t 0 φ ds, B = u, E = t u. Inserting these expressions into the remaining equation of. we finally obtain the partial differential equation to be solved σx t ux, t + x µx x ux, t. σx wx, t x ux, t σxrfx, t + s x ux, t x ux, t = 0. In order to prove ellipticity of the spatial linear second order partial differential operator in. we will use a scaling argument. Let κ R + be some parameter to be specified later in 3.4. We multiply the partial differential equation. with the nonnegative function e κt, apply the product rule, and introduce ûx, t := e κt ux, t to obtain σx tûx, t + σxκûx, t + x µx x ûx, t. σx wx, t x ûx, t σxrfx, t + se κt x ûx, t x ûx, t = 0. To derive a variational formulation which is related to. we first introduce the function space Hcurl; Ω := { v [L Ω] 3 curl v [L Ω] 3} with the graph norm v Hcurl; Ω = v L Ω + v L Ω. 4

9 Moreover, the duality pairing between the dual space [Hcurl; Ω] and Hcurl; Ω is denoted by,, while the inner product in [L Ω] 3 is denoted by,. In addition we use, Γ for the duality pairing on Γ. For the treatment of the time dependent partial differential equation., we consider the standard vector valued Lebesgue and Sobolev spaces L q 0, T ; W and H 0, T ; W := W, 0, T ; W with an appropriate Banach space W. Multiplication of the partial differential equation. with a test function v Hcurl; Ω, integration over the interior domain Ω, and integration by parts, see, e.g., [], lead to a variational formulation to find û H 0, T ; Hcurl; Ω such that for all test functions v Hcurl; Ω and almost all t 0, T there holds 0 = σ tût, v + κ σût, v + σwt ût, v µ ût µ ût, v σrft + se κt ût ût, v together with the initial condition û0 = u 0, i.e. B 0 = curl u 0.. Exterior domain n, v Γ.3 In the exterior domain Ω c, the partial differential equations. reduce to the solution of B = 0, B = 0 in Ω c, with an appropriate radiation condition. Hence we introduce the scalar potential B = Φ satisfying B = Φ = 0, B = Φ = Φ = 0 in Ω c, Φx = O as x. x Any solution of the Laplace equation in the exterior domain Ω c can be described by the representation formula Φx = U x, y [n y Φy]ds y + [n y y U x, y]φyds y for x Ω c,.4 where Γ U x, y = 4π Γ x y for x, y R 3, x y, is the related fundamental solution. From.4 we obtain a system of boundary integral equations which can be written by using the exterior Calderon projection Φ = I + K V Φ n Φ D I.5 K n Φ where for x Γ we use the single layer integral operator V Ψx = U x, yψyds y, V : H / Γ H / Γ, Γ 5

10 the double layer integral operator KΦx = [n y y U x, y]φyds y, K : H / Γ H / Γ, Γ Γ its adjoint K Ψx = [n x x U x, y]ψyds y, K : H / Γ H / Γ, and the hypersingular boundary integral operator DΦx = n x x [n y y U x, y]φyds y, D : H / Γ H / Γ. Γ The mapping properties of the boundary integral operators are well established, see, e.g., [5,, 6, ]..3 Transmission problem For the coupling of the interior partial differential equation. with the exterior problem. we need to take care of the transmission conditions.3. By using the vector potential ansatz B = u in the interior domain Ω and the scalar potential ansatz B = Φ in the exterior domain Ω c, the transmission conditions.3 read µ u n = µ 0 Φ n, u n = Φ n on Γ..6 Using the first transmission condition of.6, Stokes theorem for surface integrals [9], the first boundary integral equation in.5, and the second transmission condition of.6, we have µ u n, v = Φ n, v Γ = Φ, v n Γ Γ µ 0 µ 0 = [ V Φ n ] µ 0 I + KΦ, v n Γ = [ V [ u n] ] µ 0 I + KΦ, v n Γ. In addition, we consider the second boundary integral equation in.5, which together with the second transmission condition of.6 results in a variational formulation to find Φ H / Γ such that DΦ, Ψ Γ = I + K Φ n, Ψ Γ = I + K [ u n], Ψ Γ 6

11 is satisfied for all Ψ H / Γ. Since the hypersingular boundary integral operator D is only semi elliptic, and since the scalar potential Φ is only unique up to an additive constant, we consider the stabilized variational formulation to find Φ H / Γ such that DΦ, Ψ Γ := DΦ, Ψ Γ + Φ, Γ Ψ, Γ = I + K [ u n], Ψ Γ.7 is satisfied for all Ψ H / Γ. Although we have fixed the constant part of the scalar potential Φ when solving the modified variational problem.7, the constant part is eliminated when applying I + K. Since the stabilized hypersingular integral operator D : H / Γ H / Γ is H / Γ elliptic [0], we obtain for the unique solution of the variational formulation.7 the representation Φ = D I + K [ u n]. By using the transformations û := e κt u and Φ := e κt Φ we finally obtain for the coupling term as used in.3 µ û n, v = [V + µ 0 I + K D I + K ]T n û, T n v Γ.8 where for all v Hcurl; Ω we have, see [], Γ T n v := v n, T n v H / Γ = v n H / Γ c T v Hcurl;Ω..9 Before we introduce the variational formulation of the transmission problem..3 we will present an alternative boundary integral operator representation for the exterior problem, which is probably more suitable for the numerical implementation. As before, we write the interface term as µ u n, v = Φ, v n Γ Γ µ 0 where we now keep the first equation of.5 separately, V [n Φ] = I + KΦ, i.e. [n Φ] = V I + KΦ. Inserting this into the stabilized version of the second equation of.5 and using the second transmission condition of.6 this gives u n = n Φ = DΦ + [ I K [n Φ] = D + I K V ] I K Φ. Hence we find [ Φ = D + I K V ] I K [ u n], 7

12 and therefore we obtain the alternative representation [ µ û n, v = D + µ 0 I K V ] I K T n û, T n v Γ Unifying.8 and.0 we finally write for the coupling term µ û n, v = BT n û, T n v µ Γ, 0 Γ Γ..0 where B : H / Γ H / Γ is defined by B = V + I + K D [ I + K = D + I K V ] I K.. By using the mapping properties of all boundary integral operators we conclude Bη, η Γ c B η H / Γ, Bη H / Γ c B η H / Γ for all η H / Γ.. Now we are in a position to introduce the variational formulation of the transmission problem..3. Problem. Find û H 0, T ; Hcurl; Ω such that for all test functions v Hcurl; Ω and almost all t 0, T there holds 0 = σ tût, v + κ σût, v + µ ût, v.3 σrft σwt ût, v + se κt ût ût, v + BT n ût, T n v µ Γ 0 together with the initial condition û0 = u 0. Using the following definitions Atu, v := µ u, v σwt u, v σrft A nl u, v := + se κt u u, v, + κ σu, v, Stu, v := Atu A nl u + µ 0 T nbt n u, v.4 for all u, v Hcurl; Ω, we can reformulate Problem. in the following way: Problem. Find û H 0, T ; Hcurl; Ω such that for almost all t 0, T there holds together with the initial condition û0 = u 0. d σ + Stût = 0 dtût 8

13 3 Well posedness of the coupled formulation For the subsequent analysis, we apply the following result on nonlinear evolution problems, cf. [5, Theorem 30.A]. Theorem 3. Let V H V be an evolution triple with dim V =. Let 0 < T <, and let the following assumptions to be satisfied: a For each t 0, T, the operator St : V V is monotone and hemicontinuous. b For each t 0, T, the operator St is coercive, i.e., there exist constants M > 0 and Λ 0 such that Stv, v V V M v V Λ for all v V. c There exist a nonnegative function K L 0, T and a constant K > 0 such that Stv V K t + K v V for all v V, t 0, T. d The function t St is weakly measurable, i.e., the function t Stu, v is measurable on 0, T for all u, v V. e Let u 0 H be given. Then there exists a unique solution to the problem: Find u {v L 0, T ; V v L 0, T ; V } which fulfils d dt ut + Stut = 0 for almost all t 0, T, u0 = u 0. Although we will not give all details on the application of Theorem 3., we will sketch two important properties, Lipschitz continuity and strong monotonicity. Lemma 3. Assume ft L Ω, wt [L Ω] 3 for all t 0, T. The operator St as defined in.4 is Lipschitz continuous, i.e. for all u, u, v Hcurl; Ω and for all κ R +, t 0, T there holds Stu Stu, v c S L u u Hcurl;Ω v Hcurl;Ω 3. with { } c S L := max, σ max sup wt L µ Ω, σ max κ, cb c T, 3σ max R sup ft L min t 0,T µ Ω. 0 t 0,T 9

14 Proof. By using. and.9 we first have for all u, v Hcurl; Ω, t 0, T, [At + µ0 T nbt ] n u, v = µ u, v σwt u, v + κ σu, v + BT n u, T n v µ 0 Γ u L µ Ω v L Ω + σ max wt L Ω u L Ω v L Ω min +σ max κ u L Ω v L Ω + cb T n u µ H Γ T / n v H / Γ { } 0 max, σ max wt L µ Ω, σ max κ, cb c T u min µ Hcurl;Ω v Hcurl;Ω. 0 It remains to consider the nonlinear term A nl. For ϱ, τ R + we define The Lipschitz continuity follows from ψ ϱ τ := + ϱ τ, thus ψ ϱτ. ψ ϱ τ τ ψ ϱ τ τ τ τ for all τ, τ, ϱ R + 3. d dτ τ. + ϱ τ For s, t R + we define ϱ := se κt and set ũ i = u i to obtain u + se κt u u + se κt u, v ũ = + ϱ ũ ũ + ϱ ũ, v = ψ ϱ ũ ũ ψ ϱ ũ ũ, v [ψϱ = ψ ϱ ũ ũ ũ, v + ũ ψ ϱ ũ ] ũ, v [ = ψ ϱ ũ ũ ũ v dx + ψϱ ũ ψ ϱ ũ ] ũ v dx Ω Ω ψ ϱ ũ ũ ũ v dx + ψ ϱ ũ ψ ϱ ũ ũ v dx Ω Ω ũ ũ L Ω v L Ω + ψ ϱ ũ [ ũ ũ ] v dx Ω [ ] + ψ ϱ ũ ũ ψ ϱ ũ ũ v dx Ω 3. ũ ũ L Ω v L Ω + ũ ũ L Ω v L Ω + ũ ũ L Ω v L Ω. 0

15 By using the triangle inequality ũ ũ ũ ũ we finally obtain A nl u A nl u, v 3 σ max R ft L Ω u u L Ω v L Ω, 3.3 which together with the boundedness of the linear part concludes the proof. Let us finally consider the strong monotonicity of S. Lemma 3.3 Let St be given as in.4, and ft L Ω, wt [L Ω] 3 for all t 0, T. Let κ R + be chosen such that κ µ max σ max sup t 0,T is satisfied. Then there holds the monotonicity estimate [ wt L Ω + 3R ft L Ω] 3.4 Stu Stv, u v c M u v Hcurl;Ω for all u, v Hcurl; Ω, t 0, T 3.5 with some positive constant c M. Proof. By neglecting the positive definite operator B see., and by using the Lipschitz continuity 3.3, we have Stu Stv, u v Atu v A nl u + A nl v, u v = µ u v, u v σwt u v, u v [ ] σftr, u v t 0,T u + se κt u v + se κt v + κ σu v, u v u v L µ Ω + κ σu v L Ω max [ ] σ max sup wt L Ω + 3R ft L Ω u v L Ω σu v L Ω u v L µ Ω + κ σu v L Ω max [ wt L Ω + 3R ft L Ω σmax sup t 0,T with γ > 0. By setting ] [ γ u v L Ω + ] γ σu v L Ω γ = [ ] µ max σmax sup wt L Ω + 3R ft L Ω t 0,T

16 for we obtain κ µ max σ max Stu Stv, u v with the positive constant [ c M := min sup t 0,T µ max ; µ maxσ max σ min [ ] wt L Ω + 3R ft L Ω u v L µ Ω + κ max σu v L Ω c M u v Hcurl;Ω 3.6 [ ] ] sup t 0,T wt L Ω + 3R ft L Ω. All other assumptions of Theorem 3. follow in a similar way, so that we can conclude unique solvability of Problem.. 4 Discretisation of the coupled problem In this section we describe the spatial and temporal discretization of Problem.. First, let B some bounded and at least positive semi definite approximation of B which results from the approximate solution of the involved boundary integral equations. Specific approximations will be given in Sect Such an approximation implies an approximate operator St := At A nl + µ 0 T n BT n. Instead of Problem. we now consider a perturbed evolution equation. Problem 4. Find ũ H 0, T ; Hcurl; Ω such that for almost all t 0, T there holds σ d dtũ + Stũ = 0, ũ0 = u 0. As for S we can prove all required assumptions of Theorem 3. for the perturbed operator S, in particular, the Lipschitz continuity 3. with a Lipschitz constant c S L and the monotonicity estimate 3.5 with the same constant c M remain true. Hence we conclude the well posedness of Problem 4.. It remains to estimate the error due to the use of the approximation B. Lemma 4. Let B : H / Γ H / Γ be some bounded and positive semi definite approximation of B. Let û, ũ L 0, T ; Hcurl; Ω be the unique solutions of Problems. and 4., respectively. Then there hold the error estimates σ[ût ũt ] L Ω + c M û ũ L 0,T ;Hcurl;Ω c T c M µ 0 B BT n û L 0,T ;H / Γ

17 and Proof. σ d dt [û ũ] L 0,T ;[Hcurl;Ω] c T µ 0 From Problems. and 4. we obtain + c S L c M B BT n û L 0,T ;H / Γ. σ d dt û ũ + Atû ũ + µ 0 T n BT n [û ũ] A nl û + A nl ũ = µ 0 T n[ B B]T n û, and by using the monotonicity of S and the continuity of T n we conclude, for any γ R +, d dt σû ũ L Ω + c M û ũ Hcurl;Ω [ µ B B]T n û, T n û ũ Γ 0 µ B BT n û H Γ T / n û ũ H / Γ 0 c T µ B BT n û H Γ û ũ / Hcurl;Ω 0 [ ] c T µ 0 γ B BT n û H / Γ + γ û ũ Hcurl;Ω. In particular for γ = c M µ 0 /c T we then obtain d dt σû ũ L Ω + c M û ũ Hcurl;Ω c T c M µ B BT n û H / Γ. 0 Integration in time gives the first estimate, where we use û0 = ũ0 = u 0. On the other hand, we have For v Hcurl; Ω we then conclude σ d dt û ũ = Stũ Stû. σ d dt û ũ, v = Stũ Stû, v = Stũ Stû, v + Stû Stû, v c S L ũ û Hcurl;Ω v Hcurl;Ω + µ 0 B BT n û H / Γ T n v H / Γ. Using.9, dividing by v Hcurl;Ω, and taking the supremum over all v Hcurl; Ω, v 0, this gives σ d dt û ũ [Hcurl;Ω] c S L ũ û Hcurl;Ω + c T µ 0 B BT n û H / Γ. Integration in time and using the first estimate gives the second result. 3

18 4. Spatial discretisation Next we discuss the spatial discretization of the perturbed evolution Problem 4. where we start with the definition of appropriate discrete spaces. For the discretized vector potential in the interior domain Ω we use X h Hcurl; Ω which consists of lowest order Nédélec elements. Problem 4. Find ũ h H 0, T ; X h such that for almost all t 0, T and all v h X h there holds σ ddtũht, v h + Stũh t, v h = 0, ũ h 0 u 0, v h = 0. Theorem 4. Let ũ H 0, T ; Hcurl; Ω and ũ h H 0, T ; X h be the unique solutions of Problems 4. and 4., respectively. Let π h û X h be some approximation of the solution û Hcurl; Ω of Problem.. Then there holds the error estimate σût ũ h T L Ω + c M û ũ h L 0,T ;Hcurl;Ω [ c σût π h ût L Ω + σπ h u 0 u 0 L Ω + σu 0 ũ h 0 L Ω + π h û û L 0,T ;Hcurl;Ω + σ d dt π hû û L 0,T ;[Hcurl;Ω] + B BT ] n û L 0,T ;H / Γ. Proof. Since X h Hcurl; Ω, we first obtain the Galerkin orthogonality σ d dt ũ ũ h, v h + Stũ Stũh, v h = 0 for all v h X h, or equivalently σ d dt π hû ũ h, v h + Stπh û Stũ h, v h = Stπh û Stũ, v h + σ d dt π hû ũ, v h for all v h X h. In particular for v h = π h û ũ h we conclude, by using the monotonicity and the Lipschitz continuity of S, d dt σπ h û ũ h L Ω + c M π h û ũ h Hcurl;Ω c S L π h û ũ Hcurl;Ω π h û ũ h Hcurl;Ω + σ d dt π hû ũ [Hcurl;Ω] π h û ũ h Hcurl;Ω ] [ c S L π h û ũ Hcurl;Ω γ + γ π h û ũ h Hcurl;Ω + [ σ d ] γ dt π hû ũ [Hcurl;Ω] + γ π h û ũ h Hcurl;Ω 4

19 for some positive constants γ, γ. In particular for γ = c M/c S L and γ = c M we obtain d dt σπ h û ũ h L Ω + c M π h û ũ h Hcurl;Ω Integration in time gives [c S L ] c M π h û ũ Hcurl;Ω + c M σ d dt π hû ũ [Hcurl;Ω]. σπ h ût ũ h T L Ω + c M π h û ũ h L 0,T ;Hcurl;Ω S σπ h û0 ũ h 0 L Ω + [c L ] π h û ũ L 0,T ;Hcurl;Ω Hence we find, by using the triangle inequality, c M + c M σ d dt π hû ũ L 0,T ;[Hcurl;Ω]. σût ũ h T L Ω + c M û ũ h L 0,T ;Hcurl;Ω σût π h ût L Ω + σπ h ût ũ h T L Ω +c M û π h û L 0,T ;Hcurl;Ω + π hû ũ h L 0,T ;Hcurl;Ω σût π h ût L Ω + c M û π h û L 0,T ;Hcurl;Ω + S σπ h û0 ũ h 0 L Ω + 4[c L ] π h û ũ L 0,T ;Hcurl;Ω c M + 4 c M σ d dt π hû ũ L 0,T ;[Hcurl;Ω]. Again by using the triangle inequality we further conclude σût ũ h T L Ω + c M û ũ h L 0,T ;Hcurl;Ω σût π h ût L Ω + c M û π h û L 0,T ;Hcurl;Ω +4 σπ h û0 u 0 L Ω + 4 σu 0 ũ h 0 L Ω +8 [c S L ] S π h û û L c 0,T ;Hcurl;Ω + 8[c L ] û ũ L 0,T ;Hcurl;Ω M + 8 c M σ d dt π hû û L 0,T ;[Hcurl;Ω] + 8 c M σ d dt û ũ L 0,T ;[Hcurl;Ω] and the assertion follows by using Lemma 4.. c M 5

20 4. Time discretization We proceed with the time discretization of the semi discrete Problem 4. which was to find ũ h H 0, T ; X h such that for almost all t 0, T and all v h X h there holds σ ddtũht, v h + Stũh t, v h = 0, ũ h 0 u 0, v h = 0. For simplicity we only consider the implicit Euler scheme with a constant time step τ = T M, and with nodes t m = mτ, m = 0,..., M. If we evaluate a function u at time t m we write u m. The discretized time derivative is τ v m := τ [vm v m ], m =,..., M. Hence we obtain the following fully discretized problem. Problem 4.3 Given ũ 0 h, find for all m =,..., M ũm h X h such that σ ũm h ũm h, v h + τ St m ũ m h, v h = 0 for all v h X h. The well posedness of the fully discretized Problem 4.3 follows from the main theorem on strongly monotone operators, see, e.g., [5]. In addition we can prove the following stability estimate. Lemma 4.3 Assume ft L Ω, wt [L Ω] 3 for all t 0, T. Let κ R + be chosen according to 3.4. For the solution of the fully discrete Problem 4.3 there holds Proof. Chosing τũ m h σũ M h L Ω + τ µ max m= M ũ m h L Ω σũ 0 h L Ω. as a test function in Problem 4.3 this gives σũ m h + τat m ũ m h τa nl ũ m h + τ µ 0 T n BT n ũ m h, ũ m h = σũ m h, ũ m h, which results in σũ m h L Ω + τ ũ m h L µ Ω + κτ σũ m h L Ω max σũ m h L Ω σũ m h L Ω + τ σwt m ũ m h, ũ m h Rft m +τ σ + ϱe κtm ũ m ũm h h, ũ m h σũ m h +τ σ max sup t 0,T σũ m h + τ σ max sup t 0,T L Ω + σũ m h L Ω wt L Ω + R ft L Ω L Ω + σũ m h L Ω wt L Ω + R ft L Ω 6 ũ m h L Ω σũ m h L Ω [ γ ũ m h L Ω + ] γ σũ m h L Ω

21 for some positive constant γ. In particular for we conclude γ = µ max σmax sup wt L Ω + R ft L Ω t 0,T σũ m h L Ω + τ µ max ũ m h L Ω σũ m h L Ω [ ] +τ µ maxσ max sup wt L Ω + R ft L Ω κ σũ m h L Ω t 0,T σũ m h L Ω if we chose κ according to 3.4. Hence we obtain σũ m h L Ω + τ ũ m h L µ Ω σũ m h max and summation over m =,..., M gives the desired estimate. L Ω 4.3 Boundary element approximations In this section we will introduce suitable approximations B of the operator B as defined in.. Using the first representation in., the application of B for a given ψ H / Γ reads Bψ = V ψ + I + K D I + K ψ = V ψ + I + Kw, where w = D I + K ψ H / Γ is the unique solution of the variational problem Dw, v Γ = I + K ψ, v Γ for all v H / Γ. 4. Let Sh Γ H/ Γ be some boundary element space of, e.g., piecewise linear basis functions, which are defined with respect to an admissible and globally quasi uniform boundary element mesh. From a theoretical point of view it is not required that the boundary element mesh matches the trace of the finite element mesh, but from a practical point of view such a matching will simplify the implementation. The Galerkin discretization of the variational problem 4. is to find w h Sh Γ such that Dw h, v h Γ = I + K ψ, v h Γ for all v h S hγ. 4. 7

22 Since the stabilized hypersingular boundary integral operator D : H / Γ H / Γ is H / Γ elliptic, stability and quasi optimality follow, i.e., w h H / Γ c ψ H / Γ, w w h H / Γ c inf v h S h Γ w v h H / Γ. Hence we can define an approximate operator B by considering Bψ := V ψ + I + Kw h. 4.3 From the properties of the Galerkin approximation w h we easily conclude the boundedness of B, as well as an approximation estimate, see, e.g., [], i.e., Bψ H / Γ c ψ H / Γ, B Bψ H / Γ c inf v h S h Γ w v h H / Γ. Moreover, B : H / Γ H / Γ is H / Γ elliptic. Hence we conclude, that 4.3 defines an admissible approximation of B. However, the use of the approximation 4.3 results in the Galerkin discretization of boundary integral operators by using boundary traces of lowest order Nédélec elements. From a practical point of view, this may become rather cumbersome. Hence we are interested in a formulation, where the finite and boundary element basis functions are only linked via a generalized mass matrix. For this we consider the second representation of B as given in.. For a given ψ H / Γ, the application of B reads Bψ = [ D + I K V I K ] ψ =: w, where w H / Γ is the unique solution of the variational problem [ D + I K V I K]w, v Γ = ψ, v Γ for all v H / Γ. By introducing θ := V I Kw H / Γ this is equivalent to a variational problem to find w, θ H / Γ H / Γ such that Dw, v Γ + I K θ, v Γ = ψ, v Γ, V θ, η Γ I Kw, η Γ = is satisfied for all v, η H / Γ H / Γ. In addition to Sh Γ H/ Γ we now consider a second boundary element space Sh 0Γ H / Γ of, e.g., piecewise constant basis functions, which, for convenience, are defined with respect to the same boundary element mesh as Sh Γ. The Galerkin discretization of the variational problem 4.4 is to find w h, θ h Sh Γ S0 h Γ such that Dw h, v h Γ + I K θ h, v h Γ = ψ, v h Γ, V θ h, η h Γ I Kw h, η h Γ =

23 is satisfied for all v h, η h S h Γ S0 h Γ. The Galerkin solution w h finally implies the approximation Bψ := w h. 4.6 Using standard arguments, see, e.g., [], we can prove boundedness of B, as well as an approximation property, i.e., Bψ H / Γ c ψ H Γ, { / } B Bψ H / Γ c inf w v h H v h Sh / Γ Γ + inf θ η h H η h Sh 0 / Γ Γ. Note that B : H / Γ H / Γ may fail to be H / Γ elliptic, but B is H / Γ semi elliptic. This is due to the mixed approximation scheme as used in the definition 4.6. As in mixed finite element methods, stability would require an appropriate inf sup condition. But as seen in the proof of the monotonicity estimate 3.5, semi ellipticity of B is sufficient. Hence we conclude, that 4.6 also defines an admissible approximation. 4.4 Convergence results To present a final convergence result for the solution of the fully discrete Problem 4.3 we first cite an approximation property of the lowest order Nédélec ansatz space. Theorem 4.4 [8, Theorem 5.4] Let X h Hcurl; Ω be the space of lowest order Nédélec elements which is defined with respect to a regular and globally quasi uniform finite element mesh. For u [H Ω] 3 and u [H Ω] 3 let r h u X h be the interpolation. Then there holds the error estimate [ ] u r h u L Ω + u r h u L Ω c h u H Ω + u H Ω. 4.7 Next we define π h u X h as the Hcurl; Ω projection of u Hcurl; Ω. By using standard techniques and 4.7 we conclude the error estimate u π h u L Ω + u π h u L Ω c h u H curl;ω 4.8 for u [H Ω] 3 and u [H Ω] 3 as well as the stability estimate π h v Hcurl;Ω c π v Hcurl;Ω for all v Hcurl; Ω. 4.9 Now we are in the position to state the final convergence result. Theorem 4.5 Let û L 0, T ; Hcurl; Ω and ũ m h X h for all m =,..., M be the unique solutions of Problem. and Problem 4.3, respectively. Assume û L 0, T ; H curl; Ω L 0, T ; H curl; Ω 9

24 and d d dtû, dt û L 0, T ; [Hcurl; Ω], Then there holds the error estimate û M ũ M h L Ω + τ M m= d dtû H curl, Ω, u 0 H curl; Ω. ] û m ũ m h Hcurl;Ω cû [h + τ. Proof. The proof follows partly the lines of the proof of Theorem 4.. We will use the triangle inequality to estimate the error of an auxiliary approximation π h û m and the additional errors of the time discretization and the approximation of S on the discrete level. We start investigating the latter. We substract the variational formulation of Problem 4.3 from Problem 4. at time t = t m and obtain σ ddtũm + St m ũ m, v h = σ ũm h ũm h + τ St m ũ m h, v h for all v h X h, or equivalently σ π h û m ũ m h, v h + τ Stm π h û m St m ũ m h, v h 4.0 = σ π h û m ũ m h [, v h + σ π h û m π h û m τ d }{{} m ], v h =:A }{{} =:B +τ Stm π h û m St m π h ũ m, v h + τ Stm π h ũ m St m ũ m, v h. } {{ } } {{ } =:C =:D We set v h = ψh m := π hû m ũ m h and estimate the different terms of the error equation 4.0. The monotonicity of S gives for the left hand side terms σ π h û m ũ m h + τ Stm π h û m St m ũ m h, π h û m ũ m h The first right hand side term of 4.0 can be estimated as A = σψ m h, ψh m σψ m h L Ω + τc M ψ m h Hcurl;Ω. σψ m h L Ω σψh m L Ω σψ m h L Ω + σψh m L Ω. The Lipschitz continuity of S and the continuity 4.9 of the projection π h enable the estimate of the third right hand side term of 4.0: C = τ Stm π h û m St m π h ũ m, ψh m τc S L c π π h û m ũ m Hcurl;Ω ψh m Hcurl;Ω γ τ ψh m Hcurl;Ω + τ c S L c π û m ũ m Hcurl;Ω γ 0

25 for some positive constant γ. The Lipschitz continuity of S implies for the fourth right hand side term of 4.0, and for some positive constant γ : D = τ Stm π h ũ m St m ũ m, ψh m τc S L π h ũ m ũ m Hcurl;Ω ψh m Hcurl;Ω γ τ ψ m h Hcurl;Ω + τ c S L γ π h ũ m ũ m Hcurl;Ω. For the second right hand side term of 4.0 we obtain, for some positive constant γ 3, [ B = σ π h û m π h û m τ d ] dtũt m, ψh m τγ 3 ψh m Hcurl;Ω + γ 3 τ [π σ h û m π h û m τ d ] dtũt m [Hcurl;Ω]. We now collect the estimates of 4.0, set γ i = 3 c M, i =,..., 3, and rearrange the terms: σψh m L Ω + c Mτ ψh m Hcurl;Ω σψ m h +τ 3c S L c M π h ũ m ũ m Hcurl;Ω + 3 c M τ Summarizing over m =,..., M leads to with σπ h û M ũ M h L Ω + c Mτ L Ω + 3c S L c π τ û m ũ m Hcurl;Ω c M ] [π σ h û m π h û m τ d dtũt m M π h û m ũ m h Hcurl;Ω F m= F := σπ h û 0 ũ 0 h L Ω + 3c S L c π τ c M +τ 3c S L M π h ũ m ũ m Hcurl;Ω c M m= + 3 c M τ M û m ũ m Hcurl;Ω m= M [π σ h û m π h û m τ d ] dtũt m m= [Hcurl;Ω]. [Hcurl;Ω].

26 Inserting the approximation π h û m and application of the triangle inequality yields for the total error σû M ũ M h L Ω + c Mτ M û m ũ m h Hcurl;Ω m= σ û M π h û M L Ω + σ π h û M ũ M h +c M τ L Ω M û m π h û m Hcurl;Ω + π h û m ũ m h Hcurl;Ω m= σ û M π h û M L Ω + c Mτ M û m π h û m Hcurl;Ω + F. 4. Based on the approximation properties 4.8 of the Hcurl; Ω projection π h, the following terms of the right hand side of equation 4. can be bounded by σ û M π h û M L Ω +c Mτ m= M û m π h û m Hcurl;Ω+ σπ h û 0 ũ 0 h L Ω 4. m= ch û L 0,T ;H curl;ω + û L 0,T ;H curl;ω + u 0 H curl;ω For the last term of F we conclude, by using the triangle inequality and a Taylor expansion, requiring û to be twice continuously differentiable in time, 6 c M τ M m= [ σ π h û m π h û m τ d ] dtũt m 8 c M τ + 8 c M τ M m= +τ σ d c h d dtû +c 3 τ σ d dt [Hcurl;Ω] ] σ [π h û m π h û m û m + û m. M [ σ û m û m τ d ] dtût m [Hcurl;Ω] m= ût m ũt m dt [Hcurl;Ω] + c τ d L 0,T ;H curl;ω dt û û ũ. L 0,T ;[Hcurl;Ω] [Hcurl;Ω] 4.3 L 0,T ;[Hcurl;Ω]

27 For the last step we applied the equivalence of the discrete L norm and the L 0, T norm. In particular, we have used the mean value theorem for the first term to estimate τ = τ = τ M m= ch ] σ [π h û m π h û m û m + û m m= [Hcurl;Ω] M στ [π ûm û m h ûm û m ] τ τ M m= M m= στ [ π h d dtûξ d dtûξ ] τ d dtûξ H curl;ω. [Hcurl;Ω] For the second term we applied a Taylor expansion to obtain τ M m= [ σ û m û m τ d ] dtût m σ max τ 3 [Hcurl;Ω] [Hcurl;Ω] M d dt ûξ m. [Hcurl;Ω] Then Lemma 4. and the estimates of the BEM approximations in Subsect. 4.3 yield σ d L dt [û ũ] 0,T ;[Hcurl;Ω] m= c T µ 0 + c S L c M B BT n û L 0,T ;H / Γ c h T n û L 0,T ;Hpw / Γ c h û L 0,T ;H Ω. 4.4 The same kind of estimate holds true for the remaining part of the term F, i.e. 6 c S L c π c M τ 6 c S L c π c M M [ c π û m ũ m Hcurl;Ω + π h ũ m ũ m Hcurl;Ω] m= τ M [ c π + + c π û m ũ m Hcurl;Ω + π h û m û m Hcurl;Ω] m= c B BT n û L 0,T ;H / Γ + c π h û û L 0,T ;Hcurl;Ω 4.5 c h T n û L 0,T ;H / pw Γ + c h û L 0,T ;H curl;ω c h û L 0,T ;H curl;ω. 4.6 Now the assertion follows after summarizing all previous estimates. 3

28 5 Numerical examples In this section we present some first numerical results in order to verify the applicability of the presented FEM BEM approach. We restrict ourselves to the simplified linear model, i.e., we do not include the nonlinear saturation term A nl. For the finite element discretization of the interior problem we use hexahedral finite elements with lowest order Nédélec basis functions. This implementation was done by using the C++ library deal.ii see The boundary element approximation 4.6 was realized by using a fast multipole boundary element method []. 5. Simulation of current through a coil As a first example we consider the current through a coil which is given as the cylindrical interior domain Ω = { x R 3. < x <., x + x 3 < 0.36 } and which is decomposed into 50 hexahedral finite elements. This corresponds to 6544 degrees of freedom DOF of the finite element approximation. For the boundary element approximation we use a conforming boundary mesh with 304 triangles, i.e. 304 DOF for the flux θ h, and 54 related DOF for the potential w h, see 4.5. Figure 5.: a Hollow cylinder S grey in the interior domain Ω, b Slices for the visualisation of the finite element solution. Example In the hollow cylinder, i.e. in the coil, see Fig. 5., S = { x R < x < 0.5, 0. < x + x 3 < 0.5 }, we prescribe the velocity field w = 0, x 3, x and set w = 0 elsewhere. Moreover, we set { 000 for x Ω, 000 for x S, µx := σx := 0 else, 3 for x Ω \ S, 0 else. The initial condition is given as u 0 = 0, x 3, x. The constant time step length is chosen as τ = 0.0. For the visualisation of the finite element solution we use Tecplot 4

29 360 to calculate the rotation. This allows to visualize the magnetic field B instead of the potential u, see Fig. 5.. Fig. 5.3 shows the component B of the magnetic field at different slices in the interior domain as depicted in Fig. 5.b. Figure 5.: Magnetic field lines of the BEM solution in the plane x = 0 at timestep 40. The coil S is marked by the rectangle. Example We observe that the magnetic field points in one direction in the interior of the coil and in the opposite direction parallel to the coil. Its magnitude decreases when the distance in x direction to the coil increases. In particular, the magnetic field in the exterior behaves like a dipole field as expected. 5. Simulation of the earth s magnetic field without dynamo The second example is a first step towards a simulation of the earth s magnetic field. The interior domain Ω = B0, R is a ball with radius R = 6.38 where we consider two different meshes, see Table 5.. mesh DOF FEM finite elements DOF BEM boundary elements coarse fine Table 5.: Number of finite and boundary elements and degrees of freedom. Example We prescribe the given velocity field w = x, x, 0 in the spherical shell { } S = x R 3 < x + x + x 3 < 5 5

30 Figure 5.3: Magnetic field component B in the interior domain at slices a x =., b x =, c x = 0.65, d x = 0.5, e x = 0.3, and f x = 0. Note that the scale varies. Example and set w = 0 elsewhere. Moreover, let 07 µx := 4π and 3 05 σx := 00 for x 5, else. The initial condition is u0 = 0, x3, x >. Fig. 5.4 shows the magnetic field in a part of the exterior domain when using the fine mesh. It looks almost like a dipole field. Although an axisymmetric velocity field cannot generate an axisymmetric magnetic field see, e.g., [7] the solution shows the correct behaviour. The strength of the magnetic field decreases in time. Additionally we made another test with the coarse mesh and a 00 times larger velocity. The amplitude of the magnetic field decreases even faster in this setting cf. Fig We plot the L norm of the magnetic field in the interior domain for both velocity fields. 6 Conclusions In the present paper we have given a numerical analysis of a coupled finite and boundary element formulation to model a kinematic dynamo in R3. While the analysis is given for the nonlinear model, the numerical example only covers a simplified linear model. Although the implementation can be extended to the nonlinear model straight forward, this would 6

31 Figure 5.4: Example. a BEM solution on the fine mesh at timestep 40, b L -norm of the magnetic field on the coarse grid in the FEM domain for the given velocity field w and the velocity field 00w. One timestep is require a more efficient solution approach for the linearized model, which is beyond the scope of this paper. The model as used in this paper is based on the knowledge of the given velocity field w. A more general model will incorporate the Navier Stokes equations to describe this velocity field as an additional unknown. However, the solution of these direct problems may serve as the basis of the inverse problem, i.e., the determination of the velocity field in the interior domain from measurements on the boundary. References [] A. Buffa, P. Ciarlet Jr.: On traces for functional spaces related to Maxwell s equations. I: An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci [] Z. Chen, Q. Du, J. Zhou: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal [3] P. Ciarlet Jr., J. Zou: Fully discrete finite element approaches for time dependent Maxwell s equations. Numer. Math [4] K. H. Chan, K. Zhang, J. Zou: Spherical interface dynamos: Mathematical theory, finite element approximation, and application. SIAM J. Numer. Anal

32 [5] M. Costabel: Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal [6] P. A. Davidson: An introduction to magnetohydrodynamics. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 00. [7] J. L. Guermond, R. Laguerre, J. Leorat, C. Nore: A interior penalty Galerkin method for the MHD equations in heterogeneous domains. J. Comp. Phys [8] J. L. Guermond, R. Laguerre, J. Leorat, C. Nore: Nonlinear magnetohydrodynamics in axisymmetric heterogeneous domains using a Fourier/finite element technique and an interior penalty method. J. Comp. Phys [9] A. Giesecke, C. Norre, F. Stefani, G. Gerbeth, J. Leorat, F. Luddens, J. L. Guermond: Electromagnetic induction in non uniform domains. Geophys. Astrophys. Fluid Dynam [0] R. Hiptmair: Symmetric coupling for eddy current problems. SIAM J. Numer. Anal [] G. C. Hsiao, W. L. Wendland: Boundary Integral Equations. Springer, Heidelberg, 008. [] A. B. Iskakov, S. Descomes, J. Dormy: An integro differential formulation for magnetic induction in bounded domains: boundary element finite volume method. J. Comp. Phys [3] M. Kuhn, O. Steinbach: Symmetric coupling of finite and boundary elements for exterior magnetic field problems. Math. Methods Appl. Sci [4] R. Laguerre, C. Nore, J. Leorat, J. L. Guermond: Effects of conductivity jumps in the envelope of a kinematic dynamo model. C. R. Mecanique [5] W. Lemster: Untersuchung eines kinematischen Dynamo Problems mittels FEM BEM Kopplung. Ph.D. Thesis, Department of Mathematics and Computer Science, University of Göttingen, 0. [6] W. McLean: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, 000. [7] H. K. Moffat: Magnetic field generation in electrically conducting fluids. Cambridge University Press, Cambridge, 978. [8] P. Monk: Finite Element Methods for Maxwell s Equations. Oxford University Press, 003. [9] J. C. Nédélec: Acoustic and electromagnetic equations. vol 44 of Applied Mathematical Sciences, Springer, New York, 00. 8

33 [0] G. Of, O. Steinbach: A fast multipole boundary element method for a modified hypersingular boundary integral equation. In: Proceedings of the International Conference on Multifield Problems M. Efendiev, W. L. Wendland eds., Springer Lecture Notes in Applied Mechanics, vol., Springer, Berlin, pp , 003. [] G. Of, O. Steinbach, W. L. Wendland: The fast multipole method for the symmetric boundary integral formulation. IMA J. Numer. Anal [] O. Steinbach: Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York, 008. [3] E. P. Stephan, M. Maischak, F. Leydecker: An hp adaptive finite element/boundary element coupling method for electromagnetic problems. Comput. Mech [4] M. Xu, F. Stefani, G. Gerbeth: The integral equation method for a steady kinematic dynamo problem. J. Comp. Phys [5] E. Zeidler: Nonlinear functional analysis and its applications. II/B: Nonlinear monotone operators. Springer, New York,

34 Erschienene Preprints ab Nummer 0/ 0/ O. Steinbach, G. Unger: Convergence orders of iterative methods for nonlinear eigenvalue problems. 0/ M. Neumüller, O. Steinbach: A flexible space time discontinuous Galerkin method for parabolic initial boundary value problems. 0/3 G. Of, G. J. Rodin, O. Steinbach, M. Taus: Coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. 0/4 U. Langer, O. Steinbach, W. L. Wendland eds.: 9th Workshop on Fast Boundary Element Methods in Industrial Applications, Book of Abstracts. 0/5 A. Klawonn, O. Steinbach eds.: Söllerhaus Workshop on Domain Decomposition Methods, Book of Abstracts. 0/6 G. Of, O. Steinbach: Is the one equation coupling of finite and boundary element methods always stable? 0/ G. Of, O. Steinbach: On the ellipticity of coupled finite element and one equation boundary element methods for boundary value problems. 0/ O. Steinbach: Boundary element methods in linear elasticity: Can we avoid the symmetric formulation?