9. The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains
-
Serge Nicaise
Abstract
In this paper, we first develop a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains similar to the one for domains with smooth boundary proposed in Section 4.5.d of Costabel et al., Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth Domains, 2010. It turns out that the variational space is embedded in H1 as soon as the domain is convex. In such a case, the existence of a weak solution follows by a compact perturbation argument. As the associated boundary value problem is an elliptic system, standard shift theorem from Dauge, Elliptic Boundary Value Problems on Corner Domains - Smoothness and Asymptotics of Solutions, Springer, 1988 can be applied if the corner and edge singularities are explicitly known. We therefore describe such singularities, by adapting the general strategy from Costabel and Dauge, Arch. Ration. Mech. Anal., 151 (2000), 221-276. Finally in order to perform a wave number explicit error analysis of our problem, a stability estimate is mandatory (see Melenk and Sauter, Math. Comput., 79 (2010), 1871-1914 and Melenk and Sauter, SIAM J. Numer. Anal., 49 (2011), 1210-1243 for the Helmholtz equation). We then prove such an estimate for some particular configurations. We end up with the study of a Galerkin (h-version) finite element method using Lagrange elements and give wave number explicit error bounds in the asymptotic ranges. Some numerical tests that illustrate our theoretical results are also presented.
Abstract
In this paper, we first develop a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains similar to the one for domains with smooth boundary proposed in Section 4.5.d of Costabel et al., Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth Domains, 2010. It turns out that the variational space is embedded in H1 as soon as the domain is convex. In such a case, the existence of a weak solution follows by a compact perturbation argument. As the associated boundary value problem is an elliptic system, standard shift theorem from Dauge, Elliptic Boundary Value Problems on Corner Domains - Smoothness and Asymptotics of Solutions, Springer, 1988 can be applied if the corner and edge singularities are explicitly known. We therefore describe such singularities, by adapting the general strategy from Costabel and Dauge, Arch. Ration. Mech. Anal., 151 (2000), 221-276. Finally in order to perform a wave number explicit error analysis of our problem, a stability estimate is mandatory (see Melenk and Sauter, Math. Comput., 79 (2010), 1871-1914 and Melenk and Sauter, SIAM J. Numer. Anal., 49 (2011), 1210-1243 for the Helmholtz equation). We then prove such an estimate for some particular configurations. We end up with the study of a Galerkin (h-version) finite element method using Lagrange elements and give wave number explicit error bounds in the asymptotic ranges. Some numerical tests that illustrate our theoretical results are also presented.
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- 1. The curl–div system: theory and finite element approximation 1
- 2. Darwin and higher order approximations to Maxwell’s equations in R3 45
- 3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions 77
- 4. Numerical analysis of the half-space matching method with Robin traces on a convex polygonal scatterer 105
- 5. Eigenvalue problems in inverse electromagnetic scattering theory 145
- 6. Maxwell eigenmodes in product domains 171
- 7. Discrete regular decompositions of tetrahedral discrete 1-forms 199
- 8. Some old and some new results in inverse obstacle scattering 259
- 9. The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains 285
- 10. Time-harmonic electro-magnetic scattering in exterior weak Lipschitz domains with mixed boundary conditions 341
- 11. On an electro-magneto-elasto-dynamic transmission problem 383
- 12. Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations 403
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- 1. The curl–div system: theory and finite element approximation 1
- 2. Darwin and higher order approximations to Maxwell’s equations in R3 45
- 3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions 77
- 4. Numerical analysis of the half-space matching method with Robin traces on a convex polygonal scatterer 105
- 5. Eigenvalue problems in inverse electromagnetic scattering theory 145
- 6. Maxwell eigenmodes in product domains 171
- 7. Discrete regular decompositions of tetrahedral discrete 1-forms 199
- 8. Some old and some new results in inverse obstacle scattering 259
- 9. The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains 285
- 10. Time-harmonic electro-magnetic scattering in exterior weak Lipschitz domains with mixed boundary conditions 341
- 11. On an electro-magneto-elasto-dynamic transmission problem 383
- 12. Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations 403
