Skip to content
Licensed Unlicensed Requires Authentication Published online by De Gruyter October 6, 2022

FEM-BEM Coupling for the Maxwell–Landau–Lifshitz–Gilbert Equations via Convolution Quadrature: Weak Form and Numerical Approximation

Jan Bohn , Michael Feischl ORCID logo EMAIL logo and Balázs Kovács

Abstract

The full Maxwell equations in the unbounded three-dimensional space coupled to the Landau–Lifshitz–Gilbert equation serve as a well-tested model for ferromagnetic materials. We propose a weak formulation of the coupled system based on the boundary integral formulation of the exterior Maxwell equations. We show existence and partial uniqueness of a weak solution and propose a new numerical algorithm based on finite elements and boundary elements as spatial discretization with backward Euler and convolution quadrature for the time domain. This is the first numerical algorithm which is able to deal with the coupled system of Landau–Lifshitz–Gilbert equation and full Maxwell’s equations without any simplifications like quasi-static approximations (e.g. eddy current model) and without restrictions on the shape of the domain (e.g. convexity). We show well-posedness and convergence of the numerical algorithm under minimal assumptions on the regularity of the solution. This is particularly important as there are few regularity results available and one generally expects the solution to be non-smooth. Numerical experiments illustrate and expand on the theoretical results.

MSC 2010: 35Q61; 65M12; 65M38; 65M60

Award Identifier / Grant number: 258734477

Award Identifier / Grant number: 446431602

Funding source: Austrian Science Fund

Award Identifier / Grant number: SFB F65

Funding statement: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) “Project-ID 258734477” SFB 1173 as well as the Austrian Science Fund (FWF) under the special research program “Taming complexity in PDE systems” (grant SFB F65). The work of Balázs Kovács is additionally funded by the Heisenberg Programme of the DFG – Project-ID 446431602.

A Some Results from Operator Calculus and Auxiliary Lemmas

Lemma 23

Lemma 23 (Discrete Integration by Parts)

For N N and sequences ( a j ) j = 0 , , N , ( b j ) j = 0 , , N , it holds

[ ( t τ a ) τ + , b τ - ] [ 0 , T ] = a N b N - a 0 b 0 - [ a τ + , ( t τ b ) τ + ] [ 0 , T ] , [ a τ + , b τ - ] [ 0 , T ] = ( ( t τ ) - 1 ( a k + 1 ) k ) ( t N - 1 ) b N - [ ( ( t τ ) - 1 ( a k + 1 ) k ) τ - , ( t τ b ) τ + ] [ 0 , T ] .

Proof

It holds

[ ( t τ a ) τ + , b τ - ] [ 0 , T ] + [ a τ + , ( t τ b ) τ + ] [ 0 , T ] = τ j = 0 N - 1 a j + 1 - a j τ b j + τ j = 0 N - 1 a j + 1 b j + 1 - b j τ = j = 0 N - 1 a j + 1 b j + 1 - a j b j = a N b N - a 0 b 0 .

The second assertion can be shown similarly, by setting c 0 := 0 , c j := ( ( t τ ) - 1 ( a k + 1 ) k ) ( t j - 1 ) = τ k = 0 j - 1 a k + 1 for j = 1 , , N and using c j + 1 - c j τ = a j + 1 for j = 0 , , N - 1 :

[ a τ + , b τ - ] [ 0 , T ] + [ ( ( t τ ) - 1 ( a k + 1 ) k ) τ - , ( t τ b ) τ + ] [ 0 , T ] = τ j = 0 N - 1 c j + 1 - c j τ b j + τ j = 0 N - 1 c j + 1 b j + 1 - b j τ = c N b N - c 0 b 0 = ( ( t τ ) - 1 ( a k + 1 ) k ) ( t N - 1 ) b N .

This concludes the proof. ∎

We define for r N the exponentially weighted spaces, for c R ,

L c 2 ( [ 0 , ) , X ) := { u : [ 0 , ) X e - c u ( ) L 2 ( [ 0 , ) , X ) } ,

equipped with the norm u L c 2 ( [ 0 , ) , X ) := e - c u L 2 ( [ 0 , ) , X ) , and the exponentially weighted spaces of 𝑟-times weakly differentiable functions with zero condition at t = 0 ,

H 0 , * r ( [ 0 , ) , X ) := { ϕ : [ 0 , ) X e - c ϕ ( ) H 0 r ( [ 0 , ) , X ) for a c R } .

Using the Laplace transform ℒ and its inverse, the operator B ( t ) f (for a B : { s > σ 0 } L ( X ) and a suitable function 𝑓) is defined by

B ( t ) f := L - 1 ( B ( s ) L ( f ) ( s ) ) .

Example 24

(a) For the operator B ( s ) = s , f H 0 , * 1 ( [ 0 , ) , X ) , it holds s ( L f ) ( s ) H , and we have

B ( t ) f = t f .

Thus the Laplace differential operator t coincides with the weak derivative t if 𝑓 is weakly differentiable and f ( 0 ) = 0 .

(b) For the operator B ( s ) = s - 1 and for a function f L * 2 ( [ 0 , ) , X ) with Laplace transform s - 1 ( L f ) ( s ) H , we have

B ( t ) = t - 1 f := 0 t f ( τ ) d τ .

Thus the Laplace differential operator t - 1 coincides with the integration over time 0 t d τ .

Proof

(b) Let f L σ 2 ( [ 0 , , X ) ) ; then we have s - 1 L f ( s ) H ( max ( σ , ε ) ) for ε > 0 . Furthermore, it holds for s > max ( σ , ε ) that r 1 s e - s r f ( r ) L 1 ( [ 0 , ) , X ) , and therefore we have, by Fubini’s theorem,

L ( t - 1 f ) ( s ) = 0 e - s t 0 t f ( r ) d r d t = 0 0 1 r t e - s t f ( r ) d r d t = 0 r e - s t d t f ( r ) d r = 0 1 s e - s r f ( r ) d r = 1 s L f ( s ) .

As s - 1 L f ( s ) H , it holds t - 1 f = L - 1 s - 1 L f ( s ) .

(a) Let f H 0 , σ 1 ( [ 0 , ) , X ) for σ R . It is t - 1 t f = f , and therefore, by (b) for s max ( σ , ε ) > 0 ,

1 s L ( t f ) ( s ) = L f ( s ) .

As L ( t f ) H , it holds t f = L - 1 ( s L f ) . ∎

We want to apply the (inverse) Laplace transform to operators, B ( s ) : X X and convolute with functions f ( t ) X . The difference comparing to the scalar case is now, with the induced norm, L ( X ) is no Hilbert space, but only a Banach space. Plancherel’s formula does not hold in general.

Lemma 25

For B L 1 ( σ 0 + i R , L ( X ) ) H ( σ 0 ) the convolution with the inverse Laplace transform gives for every δ > 0 a well-defined and continuous operator

L - 1 B * : L σ 0 2 ( [ 0 , ) , X ) L σ 0 + δ 2 ( [ 0 , ) , X ) ,

and it holds

L - 1 B * u L σ 0 + δ 2 ( [ 0 , ) , X ) C ( δ ) B L 1 ( σ 0 + i R , L ( X ) ) u L σ 0 2 ( [ 0 , ) , X ) .

Proof

The proof works by combining Hölder’s inequality

L - 1 B * u L σ 0 + δ 2 ( [ 0 , ) , X ) C ( δ ) e - ( σ 0 + δ / 2 ) ( ) L - 1 B * u L ( [ 0 , ) , X )

and Young’s inequality for convolution

e - ( σ 0 + δ / 2 ) ( ) L - 1 B * u L ( [ 0 , ) , X ) e - ( σ 0 + δ / 2 ) ( ) L - 1 B L ( [ 0 , ) , L ( X ) ) e - ( σ 0 + δ / 2 ) ( ) u L 1 ( [ 0 , ) , X ) .

Then the estimates for the inverse Laplace transform which follow from the equivalence with the Fourier transform,

e - ( σ 0 + δ / 2 ) ( ) L - 1 B L ( [ 0 , ) , L ( X ) ) 1 2 π B L 1 ( σ 0 + i R , L ( X ) ) ,

and again Hölder’s inequality

e - ( σ 0 + δ / 2 ) ( ) u L 1 ( [ 0 , ) , X ) C ( δ ) u L σ 0 2 ( [ 0 , ) , X ) .

conclude the proof. ∎

Lemma 26

Lemma 26 (cf. [39, Lemma 2.1])

Let r N 0 . For

B H r := H r ( σ 0 ) := { B : { s > σ 0 } L ( X ) holomorphic B ( s ) L ( X ) C | s | r for all s > σ 0 } ,

B ( t ) f exists for every f H 0 , * r ( [ 0 , T ] , X ) and it holds L - 1 ( B ( s ) s - r L f ) H 0 , * r ( [ 0 , T ] , X ) and

B ( t ) f = 1 [ 0 , T ] L - 1 ( B ( s ) s - r L ( t r f ) )

We can define B ( t ) as a continuous operator

B ( t ) : H 0 , * r ( [ 0 , T ] , X ) L 2 ( [ 0 , T ] , X ) .

Every B H r is causal, and for every sufficiently smooth extension f ~ of 𝑓 on [ 0 , ) , it holds

B ( t ) f = 1 [ 0 , T ] L - 1 ( B ( s ) L f ~ ) .

Theorem 27

Theorem 27 (Herglotz Theorem on [ 0 , T ] , cf. [8, Lemma 2.2])

Let B , R H r ( σ 0 ) for σ 0 R , and suppose that a ( , ) : X × X C is sesquilinear and continuous. If there exists c > 0 such that, for all w X , all s > σ 0 ,

a ( w , B ( s ) w ) c R ( s ) w X 2 ,

then it holds, for all w H 0 , * r ( [ 0 , T ] , X ) , for all σ σ 0 ,

0 T e - 2 σ t a ( w ( t ) , B ( t ) w ( t ) ) d t c e - 2 σ T R ( t ) w L 2 ( [ 0 , T ] , X ) 2 .

Proof

The assertion can be shown as in the scalar case by a discrete Herglotz theorem (cf. [31, Lemma 2.1]) and the convergence of CQ.∎

Theorem 28

Theorem 28 (Discrete Herglotz Theorem on [ 0 , T ] , cf. [8, Lemma 2.1 and 2.3])

Let B H m ( σ 0 ) for σ 0 R + . For N N sufficiently large and a sequence ( w n ) n = 0 , , N X , it holds

τ j = 0 N ( B ( t τ ) w ) ( t j ) 2 C τ j = 0 N ( ( t τ ) r w ) ( t j ) 2 .

The constant 𝐶 depends on σ 0 , 𝑇 and 𝐵, but not on 𝜏.

Proof

We extend 𝑤 to a sequence ( w n ) n N such that ( ( t τ ) r w ) ( t j ) = 0 for all j > N . This is always possible by an iterative procedure, as we can write ( ( t τ ) r w ) ( t k + 1 ) = w k + 1 / τ r - f ( ( w n ) n k ) , where f ( ( w n ) n k ) does not depend on w k + 1 . Now we compute iteratively w N + 1 such that ( ( t τ ) r w ) ( t N + 1 ) = 0 , w N + 2 such that ( ( t τ ) r w ) ( t N + 2 ) = 0 , .

Now we define the finite sequence w M j := w j for j = 0 , , M and w M j = 0 , j > M . As in Lemma 15, we have, for ρ = e - 2 σ 0 τ , | ζ | < ρ and sufficiently small 𝜏,

( δ ( ζ ) τ ) 1 - e - 2 σ 0 τ τ = 0 2 σ 0 e - τ r d r 2 σ 0 e - 2 τ σ 0 > σ 0 .

With arguments similar to [8, Lemma 2.1, Lemma 2.3], we obtain

τ j = 0 e - 4 σ 0 t j ( B ( t τ ) w ) ( t j ) 2 C τ j = 0 e - 4 σ 0 t j ( ( t τ ) r w M ) ( t j ) 2 .

For j M , it is w j C t j r (this can be shown by discrete integration) and therefore

τ j = 0 e - 4 σ 0 t j ( ( t τ ) r w M ) ( t j ) 2 τ j = 0 N e - 4 σ 0 t j ( ( t τ ) r w M ) ( t j ) 2 + C ( τ , r ) e - 4 σ 0 t M t M r

and the limit M exists on the right-hand side. We obtain by discrete causality (i.e., B ( t τ ) w ( t j ) is independent of w n , n > j ), for M > N ,

τ j = 0 N e - 4 σ 0 t j ( B ( t τ ) w ) ( t j ) 2 = τ j = 0 N e - 4 σ 0 t j ( B ( t τ ) w M ) ( t j ) 2 τ j = 0 e - 4 σ 0 t j ( B ( t τ ) w M ) ( t j ) 2 .

Combining the previous estimates for the limit M gives

τ j = 0 N e - 4 σ 0 t j ( B ( t τ ) w ) ( t j ) 2 C τ j = 0 N e - 4 σ 0 t j ( ( t τ ) m w ) ( t j ) 2 .

Now the bounds e - 4 σ 0 T e - 4 σ 0 t j 1 yield the assertion. ∎

References

[1] T. Abboud, P. Joly, J. Rodríguez and I. Terrasse, Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains, J. Comput. Phys. 230 (2011), no. 15, 5877–5907. 10.1016/j.jcp.2011.03.062Search in Google Scholar

[2] G. Akrivis, M. Feischl, B. Kovács and C. Lubich, Higher-order linearly implicit full discretization of the Landau–Lifshitz–Gilbert equation, Math. Comp. 90 (2021), no. 329, 995–1038. 10.1090/mcom/3597Search in Google Scholar

[3] F. Alouges, A new finite element scheme for Landau–Lifchitz equations, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 2, 187–196. 10.3934/dcdss.2008.1.187Search in Google Scholar

[4] F. Alouges, E. Kritsikis, J. Steiner and J.-C. Toussaint, A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation, Numer. Math. 128 (2014), no. 3, 407–430. 10.1007/s00211-014-0615-3Search in Google Scholar

[5] F. Alouges and A. Soyeur, On global weak solutions for Landau–Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal. 18 (1992), no. 11, 1071–1084. 10.1016/0362-546X(92)90196-LSearch in Google Scholar

[6] L. Baňas, S. Bartels and A. Prohl, A convergent implicit finite element discretization of the Maxwell–Landau–Lifshitz–Gilbert equation, SIAM J. Numer. Anal. 46 (2008), no. 3, 1399–1422. 10.1137/070683064Search in Google Scholar

[7] Ľ. Baňas, M. Page and D. Praetorius, A convergent linear finite element scheme for the Maxwell–Landau–Lifshitz–Gilbert equations, Electron. Trans. Numer. Anal. 44 (2015), 250–270. Search in Google Scholar

[8] L. Banjai, C. Lubich and F.-J. Sayas, Stable numerical coupling of exterior and interior problems for the wave equation, Numer. Math. 129 (2015), no. 4, 611–646. 10.1007/s00211-014-0650-0Search in Google Scholar

[9] S. Bartels, J. Ko and A. Prohl, Numerical analysis of an explicit approximation scheme for the Landau–Lifshitz–Gilbert equation, Math. Comp. 77 (2008), no. 262, 773–788. 10.1090/S0025-5718-07-02079-0Search in Google Scholar

[10] S. Bartels and A. Prohl, Convergence of an implicit finite element method for the Landau–Lifshitz–Gilbert equation, SIAM J. Numer. Anal. 44 (2006), no. 4, 1405–1419. 10.1137/050631070Search in Google Scholar

[11] J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), no. 2, 185–200. 10.1006/jcph.1994.1159Search in Google Scholar

[12] J. Bohn, The Maxwell–Landau–Lifshitz–Gilbert system: Mathematical theory and numerical approximation, PhD thesis, Karlsruhe Institute of Technology (KIT), 2021. Search in Google Scholar

[13] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar

[14] A. Buffa, M. Costabel and D. Sheen, On traces for H ( curl , Ω ) in Lipschitz domains, J. Math. Anal. Appl. 276 (2002), no. 2, 845–867. 10.1016/S0022-247X(02)00455-9Search in Google Scholar

[15] A. Buffa and R. Hiptmair, Galerkin boundary element methods for electromagnetic scattering, Topics in Computational Wave Propagation, Lect. Notes Comput. Sci. Eng. 31, Springer, Berlin (2003), 83–124. 10.1007/978-3-642-55483-4_3Search in Google Scholar

[16] G. Carbou and P. Fabrie, Time average in micromagnetism, J. Differential Equations 147 (1998), no. 2, 383–409. 10.1006/jdeq.1998.3444Search in Google Scholar

[17] I. Cimrák, Existence, regularity and local uniqueness of the solutions to the Maxwell–Landau–Lifshitz system in three dimensions, J. Math. Anal. Appl. 329 (2007), no. 2, 1080–1093. 10.1016/j.jmaa.2006.06.080Search in Google Scholar

[18] I. Cimrák, A survey on the numerics and computations for the Landau–Lifshitz equation of micromagnetism, Arch. Comput. Methods Eng. 15 (2008), no. 3, 277–309. 10.1007/s11831-008-9021-2Search in Google Scholar

[19] S. Eberle, The elastic wave equation and the stable numerical coupling of its interior and exterior problems, ZAMM Z. Angew. Math. Mech. 98 (2018), no. 7, 1261–1283. 10.1002/zamm.201600236Search in Google Scholar

[20] S. Eberle, An implementation and numerical experiments of the FEM-BEM coupling for the elastodynamic wave equation in 3D, ZAMM Z. Angew. Math. Mech. 99 (2019), no. 12, Article ID e201900050. 10.1002/zamm.201900050Search in Google Scholar

[21] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651. 10.1090/S0025-5718-1977-0436612-4Search in Google Scholar

[22] M. Feischl and T. Tran, Existence of regular solutions of the Landau–Lifshitz–Gilbert equation in 3D with natural boundary conditions, SIAM J. Math. Anal. 49 (2017), no. 6, 4470–4490. 10.1137/16M1103427Search in Google Scholar

[23] M. Feischl and T. Tran, The eddy current–LLG equations: FEM-BEM coupling and a priori error estimates, SIAM J. Numer. Anal. 55 (2017), no. 4, 1786–1819. 10.1137/16M1065161Search in Google Scholar

[24] D. R. Fredkin and T. R. Koehler, Hybrid method for computing demagnetizing fields, IEEE Trans. Magn. 26 (1990), no. 2, 415–417. 10.1109/20.106342Search in Google Scholar

[25] T. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetic field, Phys. Rev. 100 (1955), 1243–1255. Search in Google Scholar

[26] J. Gorchon, C.-H. Lambert, Y. Yang, A. Pattabi, R. B. Wilson, S. Salahuddin and J. Bokor, Single shot ultrafast all optical magnetization switching of ferromagnetic co/pt multilayers, Appl. Phys. Lett. 111 (2017), no. 4, Article ID 042401. 10.1109/E3S.2017.8246170Search in Google Scholar

[27] M. J. Grote and J. B. Keller, Nonreflecting boundary conditions for time-dependent scattering, J. Comput. Phys. 127 (1996), no. 1, 52–65. 10.1006/jcph.1996.0157Search in Google Scholar

[28] T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 1999, Acta Numer. 8, Cambridge University, Cambridge (1999), 47–106. 10.1017/S0962492900002890Search in Google Scholar

[29] T. Hagstrom, A. Mar-Or and D. Givoli, High-order local absorbing conditions for the wave equation: Extensions and improvements, J. Comput. Phys. 227 (2008), no. 6, 3322–3357. 10.1016/j.jcp.2007.11.040Search in Google Scholar

[30] A. V. Kimel, Writing magnetic memory with ultrashort light pulses, Nature Rev. Mat. 4 (2019), no. 3, 2058–8437. 10.1038/s41578-019-0086-3Search in Google Scholar

[31] B. Kovács and C. Lubich, Stable and convergent fully discrete interior-exterior coupling of Maxwell’s equations, Numer. Math. 137 (2017), no. 1, 91–117. 10.1007/s00211-017-0868-8Search in Google Scholar

[32] M. Kružík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev. 48 (2006), no. 3, 439–483. 10.1137/S0036144504446187Search in Google Scholar

[33] L. Landau and E. Lifschitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z 8 (1935), 153–168. 10.1016/B978-0-08-036364-6.50008-9Search in Google Scholar

[34] K.-N. Le, M. Page, D. Praetorius and T. Tran, On a decoupled linear FEM integrator for eddy-current-LLG, Appl. Anal. 94 (2015), no. 5, 1051–1067. 10.1080/00036811.2014.916401Search in Google Scholar

[35] K.-N. Le and T. Tran, A convergent finite element approximation for the quasi-static Maxwell–Landau–Lifshitz–Gilbert equations, Comput. Math. Appl. 66 (2013), no. 8, 1389–1402. 10.1016/j.camwa.2013.08.009Search in Google Scholar

[36] A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method, Lect. Notes Comput. Sci. Eng. 84, Springer, Heidelberg, 2012. 10.1007/978-3-642-23099-8Search in Google Scholar

[37] C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129–145. 10.1007/BF01398686Search in Google Scholar

[38] C. Lubich, Convolution quadrature and discretized operational calculus. II, Numer. Math. 52 (1988), no. 4, 413–425. 10.1007/BF01462237Search in Google Scholar

[39] C. Lubich, On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer. Math. 67 (1994), no. 3, 365–389. 10.1007/s002110050033Search in Google Scholar

[40] C. Lubich, Convolution quadrature revisited, BIT 44 (2004), no. 3, 503–514. 10.1023/B:BITN.0000046813.23911.2dSearch in Google Scholar

[41] J. M. Melenk and A. Rieder, Runge–Kutta convolution quadrature and FEM-BEM coupling for the time-dependent linear Schrödinger equation, J. Integral Equations Appl. 29 (2017), no. 1, 189–250. 10.1216/JIE-2017-29-1-189Search in Google Scholar

[42] P. Monk, Finite Element Methods for Maxwell’s Equations, Numer. Math. Sci. Comput., Oxford University, New York, 2003. 10.1093/acprof:oso/9780198508885.001.0001Search in Google Scholar

[43] A. Prohl, Computational Micromagnetism, Adv. Numer. Math., B. G. Teubner, Stuttgart, 2001. 10.1007/978-3-663-09498-2Search in Google Scholar

[44] M. W. Scroggs, T. Betcke, E. Burman, W. Śmigaj and E. van ’t Wout, Software frameworks for integral equations in electromagnetic scattering based on Calderón identities, Comput. Math. Appl. 74 (2017), no. 11, 2897–2914. 10.1016/j.camwa.2017.07.049Search in Google Scholar

[45] A. Visintin, On Landau–Lifshitz’ equations for ferromagnetism, Japan J. Appl. Math. 2 (1985), no. 1, 69–84. 10.1007/BF03167039Search in Google Scholar

Received: 2022-07-13
Revised: 2022-09-05
Accepted: 2022-09-07
Published Online: 2022-10-06

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 26.11.2022 from frontend.live.degruyter.dgbricks.com/document/doi/10.1515/cmam-2022-0145/html
Scroll Up Arrow