We consider Maxwell’s equations with impedance boundary conditions on a conductive polyhedron with polyhedral holes. Well-posedness of the variational formulation is proven, a hp-discontinuous Galerkin (hp-dG) approximation as well as a priori error estimates are introduced. Next, we use the frequency as a parameter in a multi-query context. For this purpose, we derive a Reduced Basis Method (RBM) based upon the dG formulation as well as the corresponding a posteriori error bound. Numerical results indicate the efficiency and the robustness of the scheme.
Accepté le :
DOI : 10.1051/m2an/2016006
Mots clés : Maxwell’s equations, impedance, conductor, discontinuous Galerkin, Reduced Basis Method
@article{M2AN_2016__50_6_1763_0, author = {Kirchner, Kristin and Urban, Karsten and Zeeb, Oliver}, title = {Maxwell{\textquoteright}s equations for conductors with impedance boundary conditions: {Discontinuous} {Galerkin} and {Reduced} {Basis} {Methods}}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1763--1787}, publisher = {EDP-Sciences}, volume = {50}, number = {6}, year = {2016}, doi = {10.1051/m2an/2016006}, zbl = {1355.35177}, mrnumber = {3580121}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016006/} }
TY - JOUR AU - Kirchner, Kristin AU - Urban, Karsten AU - Zeeb, Oliver TI - Maxwell’s equations for conductors with impedance boundary conditions: Discontinuous Galerkin and Reduced Basis Methods JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1763 EP - 1787 VL - 50 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016006/ DO - 10.1051/m2an/2016006 LA - en ID - M2AN_2016__50_6_1763_0 ER -
%0 Journal Article %A Kirchner, Kristin %A Urban, Karsten %A Zeeb, Oliver %T Maxwell’s equations for conductors with impedance boundary conditions: Discontinuous Galerkin and Reduced Basis Methods %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1763-1787 %V 50 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016006/ %R 10.1051/m2an/2016006 %G en %F M2AN_2016__50_6_1763_0
Kirchner, Kristin; Urban, Karsten; Zeeb, Oliver. Maxwell’s equations for conductors with impedance boundary conditions: Discontinuous Galerkin and Reduced Basis Methods. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1763-1787. doi : 10.1051/m2an/2016006. http://www.numdam.org/articles/10.1051/m2an/2016006/
R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press, New York (2003). | MR | Zbl
An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput. 68 (1999) 607–631. | DOI | MR | Zbl
and ,A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23 (2001) 15–41. | DOI | MR | Zbl
, , and ,Hybrid scheduling for the parallel solution of linear systems. Parallel Computing 32 (2006) 136–156. | DOI | MR
, , and ,Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749–1779. | DOI | MR | Zbl
, , and ,A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893–1916. | DOI | MR | Zbl
and ,Convergence Rates for Greedy Algorithms in Reduced Basis Methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. | DOI | MR | Zbl
, , , , and ,On traces for in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845–867. | DOI | MR | Zbl
, and ,A. Buffa and R. Hiptmair, Galerkin boundary element methods for electromagnetic scattering. In Topics in Computational Wave Propagation. Vol. 31 of Lect. Notes Comput. Sci. Eng. Springer Berlin Heidelberg (2003) 83–124. | MR | Zbl
Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell’s problem. ESAIM: M2AN 43 (2009) 1099–1116. | DOI | Numdam | MR | Zbl
, , and ,Certified reduced basis methods and output bounds for the harmonic Maxwell’s equations. SIAM J. Sci. Comput. 32 (2010) 970–996. | DOI | MR | Zbl
, , and ,Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 9 (1975) 77–84. | Numdam | MR | Zbl
,An “” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32 (2010) 3170–3200. | DOI | MR | Zbl
, and ,An “” certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dyn. Syst. 17 (2011) 395–422. | DOI | MR | Zbl
, and ,The reduced basis method for the electric field integral equation. J. Comp. Phys. 230 (2011) 5532–5555. | DOI | MR | Zbl
, , and ,A reduced basis method for electromagnetic scattering by multiple particles in three dimensions. J. Comp. Phys. 231 (2012) 7756–7779. | DOI | MR | Zbl
, and ,Finite element analysis of a time harmonic Maxwell problem with an impedance boundary condition. IMA J. Numer. Anal. 32 (2012) 534–552. | DOI | MR | Zbl
and ,V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer (1986). | MR | Zbl
Fast Evaluation of Time-Harmonic Maxwell’s Equations Using the Reduced Basis Method. IEEE Trans. Microw. Theory Tech. 61 (2013) 2265–2274. | DOI
and ,Certified reduced basis method for the electric field integral equation. SIAM J. Sci. Comput. 34 (2012) A1777–A1799. | DOI | MR | Zbl
, and ,Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82 (2013) 247–268. | DOI | MR | Zbl
, and ,Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100 (2005) 485–518. | DOI | MR | Zbl
, , and ,A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. Acad. Sci. Paris Series I 345 (2007) 473–478. | DOI | MR | Zbl
, , and ,A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems. C. R. Math. Acad. Sci. Paris 349 (2011) 1233–1238. | DOI | MR | Zbl
, and ,W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000). | MR | Zbl
P. Monk, Finite Element Methods for Maxwell’s Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2003). | MR | Zbl
Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35 (2013) A2316 – A2340. | DOI | MR | Zbl
, , and ,The -local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput. 72 (2003) 1179–1214. | DOI | MR | Zbl
and ,Accelerated a posteriori error estimation for the reduced basis method with application to 3D electromagnetic scattering problems. SIAM J. Sci. Comput. 32 (2010) 498–520. | DOI | MR | Zbl
and ,Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics. Arch. Comput. Methods. Eng. 15 (2008) 229–275. | DOI | MR | Zbl
, and ,Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations. J. Sci. Comput. 44 (2010) 219–254. | DOI | MR | Zbl
, and ,On the constants in -finite element trace inverse inequalities. Comput. Methods Appl. Mech. Engrg. 192 (2003) 2765–2773. | DOI | MR | Zbl
and ,Cité par Sources :