A MHD problem on unbounded domains - Coupling of FEM and BEM

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1 A MHD problem on unbounded domains - Coupling of FEM and BEM Wiebke Lemster and Gert Lube Abstract We consider the MHD problem on R 3 = Ω Ω E, where Ω is a bounded, conducting Lipschitz domain and Ω E is an insulating region. After semidiscretization in time, we apply a finite element approach in Ω. A boundary element approach is used in Ω E. We present results on the well-posedness of the continuous problem and for the semidiscrete coupled problems arising within each time step. Finally we show the quasi-optimality of a regularised FE discretisation within each time step. 1 Introduction Magnetohydrodynamics MHD is the study of the flow of electrically conducting fluids in the presence of magnetic fields. In the so-called direct problem, the magnetic induction B and the electric field E are unknown. Some efforts have been made to treat this problem with Lagrange finite elements for bounded domains cf. [2, 6]. Our aim is to extend this to R 3. In the bounded, simply connected conducting domain the model is time-dependent and nonlinear. For this reason we apply finite elements there. We regularise the system with a pressure stabilisation like method to get a saddle point problem. In the unbounded part, the model reduces to a Laplace equation. Therefore, we apply a boundary element technique for this domain. We use a symmetric coupling technique cf. [15, 4] to link both methods. For more informations of symmetric coupling of FEM and BEM, see [4, 7, 9]. Properties of the used spaces and other helpful results can be found in [1, 5, 11, 12, 16]. The paper is organised as follows: In Section 2 and 3 we derive a variational formulation and Wiebke Lemster Department of Mathematics and Computer Science, University Göttingen, Germany, lemster@math.uni-goettingen.de Gert Lube Department of Mathematics and Computer Science, University Göttingen, Germany, lube@math.uni-goettingen.de 1

2 2 Wiebke Lemster and Gert Lube show the well-posedness. Section 4 deals with the time-discretised problem. Section 5 is related to the finite element approach. In Section 6 we give some final remarks. 2 Variational Formulation of the MHD Model In this section we state the problem to find a vector potential for the interior solution and a scalar potential in the exterior domain. By we denote the L 2 -norm in the conducting domain Ω which is induced by the L 2 scalar product,. We seek the interior solution in the space H 1 0,T ;H := { B L 2 0,T ;H B L 2 0,T ;H }, H := { B L 2 Ω B L 2 Ω }. Ω is divided into two disjoint parts, Ω 1 and Ω 2. We require Ω 2 = /0. We assume the known velocity w and the function f to be zero outside of Ω 2. The electric field is denoted by E the, the magnetic induction by B, the magnetic permeability by 0 < µ 1 µ µ 2 and the electric conductivity by 0 σ 1 σ σ 2. We can state the following problem: B = E, 1 t 1 µ B = σ E + w B + R f 1 + s B 2 B, 2 B = 0, 3 B = o x 1 for x. 4 We want to replace B by a potential ansatz. For the interior domain Ω we set B = u. From 1 we get E = u t + φ c. Since φ c is only unique up to a constant we choose it such that Ω φ c dx = 0. In the non-conducting region Ω E it holds that σ = 0 and µ 0 := µ is constant. Therefore 2 simplifies to B = 0. We set B = Φ in Ω E. Hence equation 3 reduces to a Laplace equation. The fundamental solution for this equation is denoted by Ux,y := 4π 1 1 x y and its normal derivative at the boundary by T x,y. We have the representation for Φ by the single-layer- and the double-layer-potential Φx = Ux,y n Φy dsy + T x,yφy dsy, x Ω E. 5 To normalise the vector potential u we use the spaces H 1 Ω := { q H 1 Ω q,1 = 0 }, H := { v H v, q = 0 q H 1 Ω }.

3 A MHD problem on unbounded domains - Coupling of FEM and BEM 3 The transmission conditions [B n] = 0 and [H n] = 0 imply the interface conditions 1 1 µ u n = Φ n and u n = Φ on. µ 0 n This leads to the following problem: Problem 1 Find u H 1 0,T ;H such that for all v H and almost all t 0,T 0 = σ u 1 1 t,v µ u, v µ u n,v σ f u R 1 + s u 2,v σw u,v. 6 In order to replace the boundary term in 6 by a formulation of the exterior problem, we use the Stokes formula on the boundary to get [ ] 1 µ u n v ds = 1µ0 ΦT n v ds with T n v := v n. If we apply this result to 5 and take the boundary values, we derive the following Calderòn equations 2V T n ux = Φx + 2KΦx, 2DΦx = T n ux 2K T n ux, where the integral operators are given by cf. [3] V ψx := K ψ x := Ux,yψy dsy, By defining the bilinear forms Dψx := n x n x Ux,yψy dsy, Kψx := T x, yψy dsy, T x, yψy dsy. A u,v := µ 1 u, v σw u,v, A t u,v := σ u t,v, K φ,v := 1 2 Id + Kφ,T nv, Dφ,ψ := Dφ,ψ + φ,1 ψ,1, V u,v := V T n u,t n v and the forms A V u,φ,v := 1 µ 0 V T n u 1 2 φ Kφ,T nv, A D u,φ,ψ := DΦ T nu + K T n u,ψ + Φ,1 ψ,1, f A nl u,v := R σ 1 + s u 2 u,v,

4 4 Wiebke Lemster and Gert Lube we get the variational problem: Problem 2 Find u,φ H 1 0,T ;H L 2 0,T ;H 1 2 such that for all v,ψ H H 1 2 and almost all t 0,T A t u + A u + A nl u + 1 µ 0 V u 1 µ 0 K Φ = 0, 7 DΦ + K u = Continuous Problem We sketch the proof of the existence and uniqueness of a solution for the continuous problem. All operators are bounded, D is invertible cf. e.g. [9] and A + A nl fullfills a Garding inequality cf. [2, 10]. Hence, the second equation can be transformed in such a way that we get from 7 an equation which only depends on u. Problem 3 Find u H 1 0,T ;H such that for almost all t 0,T Su + A t u + A nl u := A + 1 V + 1 K D 1 K u + A t u + A nl u = 0. 9 µ 0 µ 0 Therefore, we need an existence theorem for non-linear evolution problems cf. [17] Theorem 30.A.. Theorem 3.1 Let V X V be a Gelfand triple with dim V =. Assume that the operators A := S + A nl : V V fullfill the following conditions for p, q 1, with 1 p + 1 q = 1 and 0 < T < : a At is coercive for all t 0,T, i.e., there exist constants M > 0 and Λ 0 such that Atv,v M v p V Λ v V t 0,T. b At : V V is monotone and hemicontinuous for all t 0,T. c There exist a nonnegative function K 1 L q 0,T and a constant K 2 > 0 such that for all v V and t 0,T Atv V K 1 t + K 2 v p 1 V. d The function t At is weakly measurable, i.e., the function t Atu,v V is measurable for t 0,T and all u, v V. Then there exists a unique solution u W 1,p 0,T ;X for u 0 X of u t + Atut = 0, u0 = u 0. We have to modify Problem 3 in order to satisfy a coercivity condition. We multiply 9 with the function e κt, where κ is the constant for the L 2 -term in the Garding

5 A MHD problem on unbounded domains - Coupling of FEM and BEM 5 inequality which S+A nl satisfies. We denote by ũ := e κt u the new scaled potential. S is linear, so we can replace e κt Su by Sũ. To get an equivalent problem, we have to scale the constant s in A nl by e 2κt. In addition, we have e κt A t u = A t ũ + σκũ. We set u := ũ and S := S + σκid. Proof. We set p = q = 2, V = H r and X = L 2 Ω. a To show the first condition, we first note V + K D 1 K v,v 0 v H. The coercitivity follows with η = κ and c k 1 µ 2 from A nl tv,v + Stv,v 1 v 2 σ 2 u µ L Ω v v 2 σ 2 R f L Ω v v + σ 1η v 2 1 c k v 2 κc k v 2 + σ 1 η v 2 µ 2 2 { 1 min,σ 1 η κc } k v 2 H 2µ 2 2. b The monotonicity can be shown similarly. A nl is Lipschitz continuous, f Φ s Φ 1 2 f Φ s Φ 2 2,v 3 f L Ω Φ 1 Φ 2 v Φ 1,Φ 2,v H, and S is linear and bounded. Therefore both operators are hemicontinuous cf. [17] Figure c+d The third condition follows from the boundedness of the two operators S and A nl. The operator A does not depend explicitly on time. By Gronwall inequality one can derive with 2 := σ2 2 w 2 L Ω + R2 f 2 L Ω, Lt := µ 2 t σ s ds the following a-priori estimate for t [0, T ] ut u0 e Lt, u 2µ 2 t u0 e Lt. 4 Semidiscrete Problem We discretise Problem 3 in time by the implicit Euler scheme. An existence result is given. The proof relies on the main theorem of strongly monotone operators.

6 6 Wiebke Lemster and Gert Lube Theorem 4.1 Suppose X is a real Hilbert space and the operator A : X X is strongly monotone and Lipschitz continuous on X. Then, for each b X, the operator equation has a unique solution u X. Au = b For simplicity we use an equidistant partition of the time interval [0,T ] into M parts with time step size τ = M T. Hence we obtain the following semidiscrete problem: Problem 4 Find u n,φ n H H 1 2 such that for all v,ψ H H = σ un u n 1 τ 1,v + µ un, v R σ σw n u n,v + A V u n,φ n,v, 0 = A D u n,φ n,ψ. f n u n 1 + s u n 2,v For an estimate of the semidiscrete solution we need the discrete Gronwall lemma. Remark 4.2 Let {z n } Nτ n=1 z n C 1 + τc 2 be a sequence of nonnegative real numbers which fullfill z i i=0,...,n 1 for n = k,...,nτ with C 1 and C 2 independent of τ. Let z i z/k for i = 1,...,k 1. Then we obtain z n τc 2 z +C τc 2 n k for n = k,...,nτ. Hence, we get the main theorem of this section: Theorem 4.3 Problem 4 has a unique solution, if τ is chosen such that for c k < 1 µ 2 it holds c 1 τ := 1/τ κ n /2 8R 2 f n 2 L Ω /2c k > 0. Furthermore we obtain for constant σ the following estimate: max 1 n M un 2 L 2 Ω + τ n=1,...,m u n 2 L 2 Ω C u 0 2 L 2 Ω. Proof. The strong monotonicity and the boundedness of S n + A nl, defined by S n v, v := τ v,ṽ + µ v, ṽ + V + K D 1 K v,ṽ µ 0 σw n v,ṽ, R f Anl n n u,v := 1 + s u 2 u, v, can be proved by similar arguments as presented in the previous section. Therewith we can apply Theorem 4.1 to prove existence and uniqueness.

7 A MHD problem on unbounded domains - Coupling of FEM and BEM 7 5 Finite Element and Boundary Element Approach By introducing a Lagrange multiplier p n 4 can be regularised with an pressure stabilisation like term ε p n, q. Problem 5 Find ũ n, p n, Φ n H H 1 Ω H 1 2 such that for all test functions v,q,ψ H H 1 Ω H 1 2 σ u n 1,v = τ σ 1,v τ ũn + µ ũn, v + A nlũn n,v σw n ũ n,v + A V ũ n, Φ n,v + p n,v, ũ n, q = ε p n, q + p n,1q,1, 0 = A D ũ n, Φ n,ψ. One can also introduce the Lagrange multiplier p n in the original semi-discrete problem. For time steps like in Theorem 4.3 we get the error for the regularisation c 1 τ u n ũ n 2 + u n ũ n 2 + εµ 2 p n p n 2 εµ 2 p n 2. Consider the Galerkin discretisation of the coupled Problem 5. Let X h H be the lowest order edge element space see [13] and M h H 1 Ω the space of piecewise linear elements on a shape-regular tetrahedral mesh of Ω with mesh size h. The space W h H 1 2 is defined by the traces on of piecewise linear nodal elements. Problem 6 Find ũ n h, pn h, Φ n h X h M h W h such that for all test functions v,q,ψ X h M h W h σ u n 1 τ h,v = σ 1 τ ũn h,v + µ ũn h, v σw n ũ n h,v + A n nlũn h,v + A V ũ n h, Φ n h,v + pn h,v, ũ n h, q = ε pn h, q + pn h,1q,1, 0 = A D ũ n h, Φ n h,ψ. Lemma 5.1 The error between the regularised semi-discrete problem and its Galerkin formulation can be estimated as follows ũ n ũ n h H inf v h X h u n v h H + inf q h M h p q h H 1 Ω + inf ψ h W h Φ ψ h H 1 2. Proof. Defining an operator S n like S n for the new problem leads to the additional term B C 1 Bv,ṽ. The ellipticity of this operator and some other transformations lead to the statement cf. [9, 10].

8 8 Wiebke Lemster and Gert Lube 6 Conclusions We presented a symmetric coupling approach for an nonlinear MHD model cf. [2]. The continuous and the time discrete problems are well posed. A quasi-optimal estimate for the regularised problem is given. Part of our long-term objectives is to develop and implement an algorithm for this problem. Acknowledgements We want to thank O. Steinbach for stimulating discussions on the symmetric coupling of FEM and BEM. References 1. Alonso, A., Valli, A.: Some remarks on the characterization of the space of tangential traces of Hrot;Ω and the construction of an extension operator. Manuscr. Math. 89, Chan, K. H., Zhang, K., Zou, J.: Spherical Interface Dynamos: Mathematical Theory, Finite Element Approximation and Application. Technical Report CUHK Costabel, M.: Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19, Costabel, M.: Symmetric Methods for the Coupling of Finite Elements amd Boundary Elements. In: Brebbia, C.A., Wendland, W.L., Kuhn, G. eds. Boundary Elements IX Vol.1, pp Springer Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Theory and algorithms. Springer, Berlin Guermond, J.-L., Laguerre, R., Léorat, J., Nore, C.: An interior penalty Galerkin method for the MHD equations in heterogeneous domains. J. Comput. Phys. 221, Hiptmair, R.: Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40, Kress, R.: Linear integral equations. 2nd ed. Springer, New York Kuhn, M., Steinbach, O.:Symmetric coupling of finite and boundary elements for exterior magnetic field problems. Math. Methods Appl. Sci. 25, Lemster, W.: A vector potential ansatz for a MHD problem. in German Lions, J.L., Magenes, E.: Problèmes aux limites non homogenes et applications. Vol. 1. Dunod, Paris Monk, P.: Finite element methods for Maxwell s equations. Oxford University Press, Oxford Nedelec, J.-C.: Mixed Finite Elements in R 3. Numer. Math. 35, Růži cka, M.: Nonlinear functional analysis. An introduction. Nichtlineare Funktionalanalysis. Eine Einführung. Springer, Berlin Steinbach, O.: Numerical approximation methods for elliptic boundary value problems. Finite and boundary elements. Numerische Näherungsverfahren für elliptische Randwertprobleme. Finite Elemente und Randelemente. Teubner, Stuttgart Picard, R.: An elementary proof for a compact imbedding result in generalized electromagnetic theory. Math. Z. 187, Zeidler, E.: Nonlinear Functional Analysis and its Applications II/B.Springer, New York 1990