Maxwell’s equations for conductors with impedance boundary conditions: discontinuous Galerkin and reduced basis methods. (English) Zbl 1355.35177
This paper is concerned with the mathematical analysis of the Maxwell equations with impedance boundary conditions on a conductive polyhedron with polyhedral holes. The authors are interested in the well-posedness of the problem and in its variational formulation. There are established both a discontinuous Galerkin approximation and related a priori error estimates. Key features of this paper are the derivation of a reduced basis method and of a posteriori error bounds. The final part of the paper under review includes several numerical results in order to support the efficiency and the robustness of the scheme.
Reviewer: Teodora-Liliana Rădulescu (Craiova)
MSC:
| 35Q61 | Maxwell equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 78M35 | Asymptotic analysis in optics and electromagnetic theory |
| 78M30 | Variational methods applied to problems in optics and electromagnetic theory |
| 35B45 | A priori estimates in context of PDEs |
Software:
MUMPSReferences:
| [1] | R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press, New York (2003). |
| [2] | A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput.68 (1999) 607-631. · Zbl 1043.78554 · doi:10.1090/S0025-5718-99-01013-3 |
| [3] | P.R. Amestoy, I.S. Duff, J. Koster and J.-Y. L’Excellent, A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl.23 (2001) 15-41. · Zbl 0992.65018 · doi:10.1137/S0895479899358194 |
| [4] | P.R. Amestoy, A. Guermouche, J.-Y. L’Excellent and S. Pralet, Hybrid scheduling for the parallel solution of linear systems. Parallel Computing32 (2006) 136-156. · doi:10.1016/j.parco.2005.07.004 |
| [5] | D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2001) 1749-1779. · Zbl 1008.65080 · doi:10.1137/S0036142901384162 |
| [6] | C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal.35 (1998) 1893-1916. · Zbl 0913.65007 · doi:10.1137/S0036142995293766 |
| [7] | P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Convergence Rates for Greedy Algorithms in Reduced Basis Methods. SIAM J. Math. Anal.43 (2011) 1457-1472. · Zbl 1229.65193 · doi:10.1137/100795772 |
| [8] | A. Buffa, M. Costabel and D. Sheen, On traces for H(curl,{\(\Omega\)}) in Lipschitz domains. J. Math. Anal. Appl.276 (2002) 845-867. · Zbl 1106.35304 · doi:10.1016/S0022-247X(02)00455-9 |
| [9] | 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. · Zbl 1055.78013 |
| [10] | Y. Chen, J.S. Hesthaven, Y. Maday and J. Rodríguez, Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell’s problem. ESAIM: M2AN43 (2009) 1099-1116. · Zbl 1181.78019 · doi:10.1051/m2an/2009037 |
| [11] | Y. Chen, J.S. Hesthaven, Y. Maday and J. Rodríguez, Certified reduced basis methods and output bounds for the harmonic Maxwell’s equations. SIAM J. Sci. Comput.32 (2010) 970-996. · Zbl 1213.78011 · doi:10.1137/09075250X |
| [12] | P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numer.9 (1975) 77-84. · Zbl 0368.65008 |
| [13] | J.L. Eftang, A.T. Patera and E.M. Rønquist, An ”hp” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput.32 (2010) 3170-3200. · Zbl 1228.35097 · doi:10.1137/090780122 |
| [14] | J.L. Eftang, D.J. Knezevic and A.T. Patera, An ”hp” certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dyn. Syst.17 (2011) 395-422. · Zbl 1302.65223 · doi:10.1080/13873954.2011.547670 |
| [15] | M. Fares, J.S. Hesthaven, Y. Maday and B. Stamm, The reduced basis method for the electric field integral equation. J. Comp. Phys.230 (2011) 5532-5555. · Zbl 1220.78045 · doi:10.1016/j.jcp.2011.03.023 |
| [16] | M. Ganesh, J.S. Hesthaven and B. Stamm, A reduced basis method for electromagnetic scattering by multiple particles in three dimensions. J. Comp. Phys.231 (2012) 7756-7779. · Zbl 1254.78012 · doi:10.1016/j.jcp.2012.07.008 |
| [17] | G.N. Gatica and S. Meddahi, Finite element analysis of a time harmonic Maxwell problem with an impedance boundary condition. IMA J. Numer. Anal.32 (2012) 534-552. · Zbl 1243.78046 · doi:10.1093/imanum/drq041 |
| [18] | V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer (1986). · Zbl 0585.65077 |
| [19] | M.W. Hess and P. Benner, Fast Evaluation of Time-Harmonic Maxwell’s Equations Using the Reduced Basis Method. IEEE Trans. Microw. Theory Tech.61 (2013) 2265-2274. · doi:10.1109/TMTT.2013.2258167 |
| [20] | J.S. Hesthaven, B. Stamm and S. Zhang, Certified reduced basis method for the electric field integral equation. SIAM J. Sci. Comput.34 (2012) A1777-A1799. · Zbl 1247.78044 · doi:10.1137/110848268 |
| [21] | R. Hiptmair, A. Moiola and I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput.82 (2013) 247-268. · Zbl 1269.78013 · doi:10.1090/S0025-5718-2012-02627-5 |
| [22] | P. Houston, I. Perugia, A. Schneebeli and D. Schötzau, Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math.100 (2005) 485-518. · Zbl 1071.65155 · doi:10.1007/s00211-005-0604-7 |
| [23] | D.B.P. Huynh, G. Rozza, S. Sen and A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. Acad. Sci. Paris Series I345 (2007) 473-478. · Zbl 1127.65086 · doi:10.1016/j.crma.2007.09.019 |
| [24] | S. Kaulmann, M. Ohlberger and B. Haasdonk, A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems. C. R. Math. Acad. Sci. Paris349 (2011) 1233-1238. · Zbl 1269.65127 · doi:10.1016/j.crma.2011.10.024 |
| [25] | W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000). · Zbl 0948.35001 |
| [26] | P. Monk, Finite Element Methods for Maxwell’s Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2003). |
| [27] | F. Negri, G. Rozza, A. Manzoni and A. Quarteroni, Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput.35 (2013) A2316 - A2340. · Zbl 1280.49046 · doi:10.1137/120894737 |
| [28] | I. Perugia and D. Schötzau, The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput.72 (2003) 1179-1214. · Zbl 1084.78007 · doi:10.1090/S0025-5718-02-01471-0 |
| [29] | J. Pomplun and F. Schmidt, 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. · Zbl 1211.78019 · doi:10.1137/090760271 |
| [30] | G. Rozza, D.B. Phuong Huynh and A.T. Patera, 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. · Zbl 1304.65251 · doi:10.1007/s11831-008-9019-9 |
| [31] | D. Sármány, F. Izsák and J.J.W. van der Vegt, Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations. J. Sci. Comput.44 (2010) 219-254. · Zbl 1203.78046 · doi:10.1007/s10915-010-9366-1 |
| [32] | T. Warburton and J.S. Hesthaven, On the constants in hp-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Engrg.192 (2003) 2765-2773. · Zbl 1038.65116 · doi:10.1016/S0045-7825(03)00294-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
