Constructive Formulations of Resonant Maxwell's Equations

In this article we propose new constructive weak formulations for resonant time-harmonic wave equations with singular solutions. Our approach follows the limiting absorption principle and combines standard weak formulations of PDEs with properties of elementary special functions adapted to the singularity of the solutions, called manufactured solutions. We show the well-posedness of several formulations obtained by these means for the limit problem in dimension one, and propose a generalization in dimension two.

[1]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

[2]  Lucas Chesnel,et al.  T-COERCIVITY FOR SCALAR INTERFACE PROBLEMS BETWEEN DIELECTRICS AND METAMATERIALS , 2011 .

[3]  Ralf Hiptmair,et al.  STABILITY RESULTS FOR THE TIME-HARMONIC MAXWELL EQUATIONS WITH IMPEDANCE BOUNDARY CONDITIONS , 2011 .

[4]  Patrick Ciarlet,et al.  T-coercivity: Application to the discretization of Helmholtz-like problems , 2012, Comput. Math. Appl..

[5]  Lucas Chesnel,et al.  RADIATION CONDITION FOR A NON-SMOOTH INTERFACE BETWEEN A DIELECTRIC AND A METAMATERIAL , 2013 .

[6]  Lucas Chesnel,et al.  A curious instability phenomenon for a rounded corner in presence of a negative material , 2013, Asymptot. Anal..

[7]  Franck Assous,et al.  Time-dependent Maxwell's equations with charges in singular geometries , 2006 .

[8]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[9]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[10]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[11]  Francis F. Chen,et al.  Amplification and absorption of electromagnetic waves in overdense plasmas , 1974 .

[12]  Bruno Després,et al.  Weak solutions to Friedrichs systems with convex constraints , 2011 .

[13]  B. Després,et al.  Singular solutions for the plasma at the resonance , 2017 .

[14]  Ricardo Weder,et al.  Hybrid resonance of Maxwell's equations in slab geometry , 2012, 1210.0779.

[15]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[16]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[17]  Serge Nicaise,et al.  Corner Singularities of Maxwell Interface and Eddy Current Problems , 2004 .

[18]  Howard H. C. Chang,et al.  Radio waves in the ionosphere , 1962 .

[19]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[20]  J. Roche,et al.  Electromagnetic wave propagation and absorption in magnetised plasmas: variational formulations and domain decomposition , 2015, 1502.00954.

[21]  Hoai-Minh Nguyen,et al.  Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients , 2015, 1507.01730.