Hybridized DG methods

!!!Related publications

%lfloat text-align=center width=700px% http://www-sop.inria.fr/nachos/pics/results/hdg/hdg_conv.png | Propagation of a plane wave in vacuum (2d case) : asymptotic convergence with non-uniform triangular meshes.\\\

%newwin% [[http://dx.doi.org/10.1016/j.cpc.2015.02.017 | Comput. Phys. Comm. (2015)]]\\

This study is conducted in collaboration with Jay Gopalakrishnan (), Liang Li (), Nicole Olivares () and Ronan Perrussel ().\\

%newwin% [[http://arxiv.org/abs/1503.06223 | Available on arXiv]]\\

HDG methods  introduce an additional  ''hybrid'' variable on the  faces of the  elements, on  which the  definition of  the  local (element-wise) solutions is based.  A  so-called ''conservativity condition'' is imposed on the numerical trace, whose definition involved the hybrid variable, at  the interface  between  neighboring elements.  As  a result,  HDG methods produce a linear system in terms of the DOFs of the additional hybrid variable  only.  In  this way, the  number of  globally coupled DOFs is reduced.  The local values of the  electromagnetic fields can be  obtained by  solving local  problems element-by-element.  In this work,  for the  system  of 3d  time-harmonic  Maxwell's equations,  we propose  a HDG  formulation  taking the  tangential  component of  the magnetic  field as  the hybrid  variable.  We show  that the  reduced system  of    the  hybrid  variable    has  a  wave-equation-like characterization,  and  the  tangential  components of  the  numerical traces  for  both  electric  and magnetic  fields  are  single-valued. Moreover,  numerical results  seems to  indicate that  the approximate solutions for both '''E''' and '''H''' have optimal convergence orders.\\\

HDG methods  introduce an additional  ''hybrid'' variable on the  faces of the  elements, on  which the  definition of  the  local (element-wise) solutions is based.  A  so-called ''conservativity condition'' is imposed on the numerical trace, whose definition involved the hybrid variable, at  the interface  between  neighboring elements.  As  a result,  HDG methods produce a linear system in terms of the DOFs of the additional hybrid variable  only.  In  this way, the  number of  globally coupled DOFs is reduced.  The local values of the  electromagnetic fields can be  obtained by  solving local  problems element-by-element.  In this work,  for the  system  of 3d  time-harmonic  Maxwell's equations,  we propose  a HDG  formulation  taking the  tangential  component of  the magnetic  field as  the hybrid  variable.  We show  that the  reduced system  of    the  hybrid  variable    has  a  wave-equation-like characterization,  and  the  tangential  components of  the  numerical traces  for  both  electric  and magnetic  fields  are  single-valued. Moreover,  numerical results  seems to  indicate that  the approximate solutions for both '''E''' and '''H''' have optimal convergence orders.

%lfloat text-align=center width=700px% http://www-sop.inria.fr/nachos/pics/results/hdg/hdg_conv.png | Propagation of a plane wave in vacuum (2d case) : asymptotic convergence with non-uniform triangular meshes.\\

%lfloat text-align=center width=700px% http://www-sop.inria.fr/nachos/pics/results/hdg/hdg_conv.png | Propagation of a plane wave in vacuum (2d case) : asymptotic convergence with non-uniform triangular meshes.

%lfloat text-align=center width=750px% http://www-sop.inria.fr/nachos/pics/results/hdg/hdg_conv.png | Propagation of a plane wave in vacuum (2d case) : asymptotic convergence with non-uniform triangular meshes.

%lfloat text-align=center width=700px% http://www-sop.inria.fr/nachos/pics/results/hdg/hdg_conv.png | Propagation of a plane wave in vacuum (2d case) : asymptotic convergence with non-uniform triangular meshes.

%lfloat text-align=center width=700px% http://www-sop.inria.fr/nachos/pics/results/hdg/hdg_conv.png | Propagation of a plane wave in vacuum (2d case). Asymptotic convergence with non-uniform triangular meshes.

%lfloat text-align=center width=650px% http://www-sop.inria.fr/nachos/pics/results/hdg/hdg_conv.png | Propagation of a plane wave in vacuum (2d case). Asymptotic convergence with non-uniform triangular meshes.

%lfloat text-align=center width=650px% http://www-sop.inria.fr/nachos/pics/results/hdg/pw_conv_ug.jpg | Propagation of a plane wave in vacuum (2d case). Asymptotic convergence with non-uniform triangular meshes.

%lfloat text-align=center width=550px% http://www-sop.inria.fr/nachos/pics/results/hdg/pw_conv_ug.jpg | Propagation of a plane wave in vacuum (2d case). Asymptotic convergence with non-uniform triangular meshes.

%lfloat text-align=center width=250px% http://www-sop.inria.fr/nachos/pics/results/hdg/pw_conv_ug.jpg | Propagation of a plane wave in vacuum (2d case). Asymptotic convergence with non-uniform triangular meshes.

Discontinuous Galerkin  (DG) methods have  been extensively considered for obtaining approximate solution  of Maxwell's equations.  Thanks to the discontinuity of the approximation,  this kind of methods has many advantages, such  as adaptivity to complex geometries  through the use of unstructured  possibly non-conforming meshes,  easily obtained high order  accuracy,  hp-adaptivity  and  natural  parallelism.  However, despite  these  advantages,  DG  methods  have  one  main  drawback particularly sensitive for (psuedo)-stationary problems: the number of globally coupled  degrees of freedom  (DOFs) is much greater  than the number of DOFs  required by conforming finite element  methods for the same  accuracy.  Consequently, DG  methods are  expensive in  terms of both  CPU time  and memory  consumption, especially  for time-harmonic problems.  Hybridization of  DG methods  is devoted  to  address this issue while keeping  all the advantages of DG  methods.  The design of such  a  hybridizable  discontinuous  Galerkin (HDG)  method  for  the discretization of  the system of 3d  time-harmonic Maxwell's equations is considered here.\\\

solutions for both '''E''' and '''H''' have optimal convergence orders.