Mesh requirements for the finite element approximation of problems with sign-changing coefficients

Transmission problems with sign-changing coefficients occur in electromagnetic theory in the presence of negative materials surrounded by classical materials. For general geometries, establishing Fredholmness of these transmission problems is well-understood thanks to the $$\mathtt {T}$$T-coercivity approach. Moreover, for a plane interface, there exist meshing rules that guarantee an optimal convergence rate for the finite element approximation. We propose here a new treatment at the corners of the interface which allows to design meshing rules for an arbitrary polygonal interface and then recover standard error estimates. This treatment relies on the use of simple geometrical transforms to define the meshes. Numerical results illustrate the importance of this new design.

[1]  David R. Smith,et al.  Metamaterials and Negative Refractive Index , 2004, Science.

[2]  Lucas Chesnel,et al.  On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients , 2015, J. Comput. Phys..

[3]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[4]  K. Ramdani,et al.  Lignes supraconductrices : analyse mathematique et numerique , 1999 .

[5]  Lucas Chesnel,et al.  Étude de quelques problèmes de transmission avec changement de signe. Application aux métamatériaux. , 2012 .

[6]  Martin Vohralík,et al.  Robust a posteriori error control for transmission problems with sign-changing coefficients using localization of dual norms , 2015 .

[7]  K. Ramdani,et al.  Analyse spectrale et singularits d'un problme de transmission non coercif , 1999 .

[8]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[9]  Patrick Ciarlet,et al.  Time harmonic wave diffraction problems in materials with sign-shifting coefficients , 2010, J. Comput. Appl. Math..

[10]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

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

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

[13]  A. H. Schatz,et al.  An observation concerning Ritz-Galerkin methods with indefinite bilinear forms , 1974 .

[14]  Lucas Chesnel,et al.  Eigenvalue problems with sign-changing coefficients , 2017 .

[15]  Lucas Chesnel,et al.  T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients , 2013, Numerische Mathematik.

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

[17]  Eric T. Chung,et al.  A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials , 2013, J. Comput. Appl. Math..

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

[19]  Camille Carvalho,et al.  Étude mathématique et numérique de structures plasmoniques avec coins , 2015 .

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

[21]  S. Ramakrishna,et al.  Physics of negative refractive index materials , 2005 .

[22]  Martin Costabel,et al.  A direct boundary integral equation method for transmission problems , 1985 .

[23]  Serge Nicaise,et al.  A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients , 2010, J. Comput. Appl. Math..