2. Darwin and higher order approximations to Maxwell’s equations in R3
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Sebastian Bauer
Abstract
This contribution is concerned with an asymptotic expansion of Maxwell’s equations in case that charge velocities are small in comparison with the speed of light. In every order of expansion, two curl-div systems have to be solved in which solutions of the previous order enter on the right-hand side. It is proved that in case of a bounded underlying domain Ω in every order k of expansion solutions are well-defined and give an approximation of solutions of Maxwell’s equations with a L2 error bound O((v/c)k+1) if initial values of the electromagnetic fields are suitably adapted. In case of Ω = ℝ3, weighted L2 spaces are used for solving curl-div systems. It is shown that solutions of the approximation are only L2, if certain derivatives of the multipole expansion of the sources vanish. For that reason, a careful analysis of mapping properties of vector differential operators in weighted L2 spaces is given which might be of interest in its own right.
Abstract
This contribution is concerned with an asymptotic expansion of Maxwell’s equations in case that charge velocities are small in comparison with the speed of light. In every order of expansion, two curl-div systems have to be solved in which solutions of the previous order enter on the right-hand side. It is proved that in case of a bounded underlying domain Ω in every order k of expansion solutions are well-defined and give an approximation of solutions of Maxwell’s equations with a L2 error bound O((v/c)k+1) if initial values of the electromagnetic fields are suitably adapted. In case of Ω = ℝ3, weighted L2 spaces are used for solving curl-div systems. It is shown that solutions of the approximation are only L2, if certain derivatives of the multipole expansion of the sources vanish. For that reason, a careful analysis of mapping properties of vector differential operators in weighted L2 spaces is given which might be of interest in its own right.
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
