Weighted regularization for composite materials in electromagnetism

Patrick Ciarlet Jr.; François Lefèvre; Stéphanie Lohrengel; Serge Nicaise

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 1, page 75-108
  • ISSN: 0764-583X

Abstract

top
In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace of ( 𝐜𝐮𝐫𝐥 ;Ω) whose fields 𝐮 satisfy w α div ( ε 𝐮 )∈L2(Ω) and have vanishing tangential trace or tangential trace in L2( Ω ). The weight function w ( 𝐱 ) is equivalent to the distance of 𝐱 to the geometric singularities and the minimal weight parameter α is given in terms of the singular exponents of a scalar transmission problem. A density result is proven that guarantees the approximability of the solution field by piecewise regular fields. Numerical results for the discretization of the source problem by means of Lagrange Finite Elements of type P1 and P2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed.

How to cite

top

Ciarlet Jr., Patrick, et al. "Weighted regularization for composite materials in electromagnetism." ESAIM: Mathematical Modelling and Numerical Analysis 44.1 (2010): 75-108. <http://eudml.org/doc/250791>.

@article{CiarletJr2010,
abstract = { In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace of $\{\cal H\}$($\{\bf curl\}$;Ω) whose fields $\textit\{\textbf\{ u\}\}$ satisfy $w^\alpha$ div ($\varepsilon\{\textit\{\textbf\{u\}\}\}$)∈L2(Ω) and have vanishing tangential trace or tangential trace in L2($\partial\Omega$). The weight function $w(\bf x)$ is equivalent to the distance of $\bf x$ to the geometric singularities and the minimal weight parameter α is given in terms of the singular exponents of a scalar transmission problem. A density result is proven that guarantees the approximability of the solution field by piecewise regular fields. Numerical results for the discretization of the source problem by means of Lagrange Finite Elements of type P1 and P2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed. },
author = {Ciarlet Jr., Patrick, Lefèvre, François, Lohrengel, Stéphanie, Nicaise, Serge},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Maxwell's equations; interface problem; singularities of solutions; density results; weighted regularization},
language = {eng},
month = {3},
number = {1},
pages = {75-108},
publisher = {EDP Sciences},
title = {Weighted regularization for composite materials in electromagnetism},
url = {http://eudml.org/doc/250791},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Ciarlet Jr., Patrick
AU - Lefèvre, François
AU - Lohrengel, Stéphanie
AU - Nicaise, Serge
TI - Weighted regularization for composite materials in electromagnetism
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 1
SP - 75
EP - 108
AB - In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace of ${\cal H}$(${\bf curl}$;Ω) whose fields $\textit{\textbf{ u}}$ satisfy $w^\alpha$ div ($\varepsilon{\textit{\textbf{u}}}$)∈L2(Ω) and have vanishing tangential trace or tangential trace in L2($\partial\Omega$). The weight function $w(\bf x)$ is equivalent to the distance of $\bf x$ to the geometric singularities and the minimal weight parameter α is given in terms of the singular exponents of a scalar transmission problem. A density result is proven that guarantees the approximability of the solution field by piecewise regular fields. Numerical results for the discretization of the source problem by means of Lagrange Finite Elements of type P1 and P2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed.
LA - eng
KW - Maxwell's equations; interface problem; singularities of solutions; density results; weighted regularization
UR - http://eudml.org/doc/250791
ER -

References

top
  1. C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Meth. Appl. Sci.21 (1998) 823–864.  
  2. F. Assous, P. Degond, E. Heintzé, P.-A. Raviart and J. Segré, On a finite element method for solving the three-dimensional Maxwell equations. J. Comput. Phys.109 (1993) 222–237.  
  3. F. Assous, P. Degond and J. Segré, Numerical approximation of the Maxwell equations in inhomogeneous media by a P1 conforming finite element method. J. Comput. Phys.128 (1996) 363–380.  
  4. F. Assous, P. Ciarlet Jr. and E. Sonnendrücker, Resolution of the Maxwell equations in a domain with reentrant corners. Math. Mod. Num. Anal.32 (1998) 359–389.  
  5. F. Assous, P. Ciarlet Jr., P.-A. Raviart and E. Sonnendrücker, A characterization of the singular part of the solution to Maxwell's equations in a polyhedral domain. Math. Meth. Appl. Sci.22 (1999) 485–499.  
  6. F. Assous, P. Ciarlet Jr. and J. Segré, Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method. J. Comput. Phys.161 (2000) 218–249.  
  7. M. Birman and M. Solomyak, L2-theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv.42 (1987) 75–96.  
  8. M. Birman and M. Solomyak, On the main singularities of the electric component of the electro-magnetic field in regions with screens. St. Petersbg. Math. J.5 (1993) 125–139.  
  9. D. Boffi, F. Brezzi and L. Gastaldi, On the convergence of eigenvalues for mixed formulations. Annali Sc. Norm. Sup. Pisa Cl. Sci.25 (1997) 131–154.  
  10. A.-S. Bonnet-Ben Dhia, C. Hazard and S. Lohrengel, A singular field method for the solution of Maxwell's equations in polyhedral domains. SIAM J. Appl. Math.59 (1999) 2028–2044.  
  11. A. Buffa, P. Ciarlet Jr. and E. Jamelot, Solving electromagnetic eigenvalue problems in polyhedral domains. Numer. Math.113 (2009) 497–518.  
  12. P. Ciarlet Jr., Augmented formulations for solving Maxwell equations. Comp. Meth. Appl. Mech. Eng.194 (2005) 559–586.  
  13. P. Ciarlet Jr. and G. Hechme, Computing electromagnetic eigenmodes with continuous Galerkin approximations. Comp. Meth. Appl. Mech. Eng.198 (2008) 358–365.  
  14. P. Ciarlet Jr. and G. Hechme, Mixed, augmented variational formulations for Maxwell's equations: Numerical analysis via the macroelement technique. Numer. Math. (Submitted).  
  15. P. Ciarlet Jr., C. Hazard and S. Lohrengel, Les équations de Maxwell dans un polyèdre : un résultat de densité. C. R. Acad. Sci. Paris, Ser. I326 (1998) 1305–1310.  
  16. M. Costabel and M. Dauge, Un résultat de densité pour les équations de Maxwell régularisées dans un domaine lipschitzien. C. R. Acad. Sci. Paris, Ser. I327 (1998) 849–854.  
  17. M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Rational Mech. Anal.151 (2000) 221–276.  
  18. M. Costabel and M. Dauge, Weighted regularization of Maxwell's equations in polyhedral domains. Numer. Math.93 (2002) 239–277.  
  19. M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems. ESAIM: M2AN33 (1999) 627–649.  
  20. M. Dauge, Benchmark computations for Maxwell equations for the approximation of highly singular solutions. (2004). See Monique Dauge's personal web page at the location  URIhttp://perso.univ-rennes1.fr/monique.dauge/core/index.html
  21. P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Meth. Appl. Sci.7 (1997) 957–991.  
  22. P. Grisvard, Edge behaviour of the solution of an elliptic problem. Math. Nachr.132 (1987) 281–299.  
  23. P. Grisvard, Singularities in boundary value problems, RMA22. Masson (1992).  
  24. C. Hazard and M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell's equations. SIAM J. Math. Anal.27 (1996) 1597–1630.  
  25. C. Hazard and S. Lohrengel, A singular field method for Maxwell's equations: numerical aspects for 2D magnetostatics. SIAM J. Numer. Anal.40 (2002) 1021–1040.  
  26. B. Heinrich, S. Nicaise and B. Weber, Elliptic interface problems in axisymmetric domains. I: Singular functions of non-tensorial type. Math. Nachr.186 (1997) 147–165.  
  27. D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity, RMA5. Masson (1987).  
  28. S. Lohrengel and S. Nicaise, Singularities and density problems for composite materials in electromagnetism. Comm. Partial Diff. Eq.27 (2002) 1575–1623.  
  29. J.M.-S. Lubuma and S. Nicaise, Dirichlet problems in polyhedral domains. I: Regularity of the solutions. Math. Nachr.168 (1994) 243–261.  
  30. P. Monk, Finite element methods for Maxwell's equations. Oxford University Press, UK (2003).  
  31. M. Moussaoui, H ( div , rot , Ω ) dans un polygone plan. C. R. Acad. Sci. Paris, Ser. I322 (1996) 225–229.  
  32. S. Nazarov and B. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, Exposition in Mathematics13. De Gruyter, Berlin, Germany (1994).  
  33. S. Nicaise, Polygonal interface problems. Peter Lang, Berlin, Germany (1993).  
  34. S. Nicaise and A.-M. Sändig, General interface problems I, II. Math. Meth. Appl. Sci.17 (1994) 395–450.  
  35. B. Smith, P. Bjorstad and W. Gropp, Domain decomposition. Parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, New York, USA (1996).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.