Existence and uniqueness of a solution for a field/circuit coupled problem
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 1045-1061.

In this paper we show unique solvability of an abstract coupled problem which originates from a field/circuit coupled problem. The coupled problem arises in particular from modified nodal analysis equations linked with an eddy current problem via solid conductor model. The proof technique in the paper relies on Rothe’s method and the theory of monotone operator. We also provide error estimates for time discretization.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016052
Classification : 47N20, 65M12, 65M20
Mots clés : Coupled problem, field/circuit, uniqueness, convergence, time discretization
Slodička, Marian 1 ; Vrábel’, Vladimír 1

1 Department of Mathematical Analysis, Research group of Numerical Analysis and Mathematical Modeling (NaM2), Ghent University, Ghent, Belgium.
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Slodička, Marian; Vrábel’, Vladimír. Existence and uniqueness of a solution for a field/circuit coupled problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 1045-1061. doi : 10.1051/m2an/2016052. http://www.numdam.org/articles/10.1051/m2an/2016052/

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. | DOI | MR | Zbl

S. Baumanns, Coupled Electromagnetic Field/Circuit Simulation. Modeling and Numerical Analysis. Logos Verlag Berlin GmbH (2012).

F. Bachinger, U. Langer and J. Schöberl, Numerical analysis of nonlinear multiharmonic eddy current problems. Numer. Math. 100 (2005) 593–616. | DOI | MR | Zbl

D.D. Bainov and P.S. Simeonov, Integral inequalities and applications. Vol. 57 of Mathematics and Its Applications. Springer (1992). | MR | Zbl

A. Bermúdez, B. López-Rodríguez, R. Rodríguez and P. Salgado, An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions. ESAIM: M2AN 47 (2013) 875–902. | DOI | Numdam | MR | Zbl

D. Estévez Schwarz and C. Tischendorf, Structural analysis of electric circuits and consequences for MNA. Int. J. Circuit Theory Appl. 28 (2000) 131–162. | DOI | Zbl

J.D. Jackson and J.D. Jackson, Classical electrodynamics. Vol. 3. Wiley New York (1962). | MR | Zbl

J.G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. part iv: Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353–384. | DOI | MR | Zbl

R. Hiptmair and O. Sterz, Current and voltage excitations for the eddy current model. Int. J. Numer. Model. Electron. Netw., Devices Fields 18 (2005) 1–21. | DOI | Zbl

A. Kufner, O. John and S. Fuıč´k, Function spaces. Vol. 3. Springer (1977). | MR

M. Kolmbauer, Existence and uniqueness of eddy current problems in bounded and unbounded domains. Technical Report NuMa-Report 2011-03, Johannes Kepler University Linz, Institute of Computational Mathematics. Linz (2011).

M. Matthes, Numerical Analysis of Nonlinear Partial Differential-Algebraic Equations: A Coupled and an Abstract Systems Approach, Logos Verlag Berlin GmbH (2012).

J. Nečas, Introduction to the theory of nonlinear elliptic equations. Chichester: John Wiley and Sons Ltd (1986). | MR | Zbl

A. Rodriguez and A. Valli, Voltage and current excitation for time-harmonic eddy-current problems. SIAM J. Appl. Math. 68 (2008) 1477–1494. | DOI | MR | Zbl

J. Kačur, Method of Rothe in evolution equations, Vol. 80 of Teubner Texte zur Mathematik. Teubner, Leipzig (1985). | MR | Zbl

T. Roubíček, Nonlinear Partial Differential Equations with Applications. Birkhäuser, Berlin (2005). | MR | Zbl

M. Slodička, A time discretization scheme for a nonlinear degenerate eddy current model for ferromagnetic materials. IMAJNA 26 (2006) 173–187. | MR | Zbl

S. Schöps, Multiscale modeling and multirate time-integration of field/circuit coupled problems. Ph.D. thesis, Katholieke Universiteit Leuven (2011).

S. Schöps, H. De Gersem and T. Weiland, Winding functions in transient magnetoquasistatic field-circuit coupled simulations. COMPEL: Int. J. Comput. Math. Electr. Electron. Eng. 32 (2013) 2063–2083. | DOI | MR | Zbl

C. Tischendorf, Coupled systems of differential algebraic and partial differential equations in circuit and device simulation. Habilitationsschrift, Humboldt-University Berlin (2003).

M. Vajnberg, Variational method and method of monotone operators in the theory of nonlinear equations. John Wiley and Sons (1973). | MR | Zbl

E. Zeidler, Nonlinear Functional Analysis and its Applications II/A: Linear Monotone Operators. Springer, New York (1990). | Zbl

E. Zeidler, Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators, Springer. New York (1990). | MR | Zbl

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