Locally implicit DGTD methods for electromagnetics

Propagation of a standing wave in  a cubic PEC cavity. Examples of two
locally  refined  meshes (implicit treatment:  ''red regions'').  From top  to bottom: interior skeletons  of 3D meshes and  cross sections of the  3D meshes resulting (on the  left: 635 vertices and 2,968  tetraedra; on the right: 7,759 vertices and 40,616 tetraedra).

Propagation  of  a standing  wave  in  a  cubic PEC  cavity:  numerical convergence and  maximum error ( L'_2_'-norm) in function  of final CPU time for  the locally implicit  and fully  explicit DGTD-P'_k_' methods (top - bottom, respectively).

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Propagation  of  a standing  wave  in  a  cubic PEC  cavity:  numerical
convergence and  maximum error ( L'_2_'-norm) in function  of final CPU
time for  the locally implicit  and fully  explicit DGTD-P'_k_' methods
(top - bottom, respectively).

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%newwin% [[h/tel.archives-ouvertes.fr/tel-00950386 | Doctoral thesis]], University of Nice - Sophia Antipolis (2013)

%center% Propagation of a standing wave in  a cubic PEC cavity. Examples of two

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The  use of  unstructured  meshes  (based on  triangles  in two  space dimensions and tetrahedra  in three space dimensions)  is an important feature of  DGTD methods  studied in the team  for dealing  with complex  geometries and heterogeneous propagation  media.  Moreover,  discontinuous Galerkin  discretization methods  are  naturally  adapted  to  local,  conforming  as  well  as non-conforming,  refinement  of  the  underlying  mesh,  Most  of  the existing DGTD  methods rely on  explicit time integration  schemes and lead to block diagonal mass matrices  which is often recognized as one of  the main  advantages  with regards  to  continuous finite  element methods.  However,  explicit DGTD  methods are  also constrained  by a stability condition  that can  be very  restrictive on  highly refined meshes  and  when  the  local  approximation  relies  on  high  order polynomial  interpolation.

There are  basically three strategies  that can be considered  to mitigate this computational  efficiency problem.  A first  approach  is to use  an unconditionally stable implicit  time integration scheme to overcome  the restrictive constraint  on the time step  for locally refined meshes. A second approach is to use  a local time  stepping strategy combined with an explicit time  integration scheme.  In the  third approach, the time  step size restriction is overcome by using a hybrid explicit-implicit procedure. In the  case, one blends a  time implicit and a  time explicit schemes where only the solution variables defined on the smallest elements are treated implicitly.

The  use of  unstructured  meshes  (based on  triangles  in two  space dimensions and tetrahedra  in three space dimensions)  is an important feature of  DGTD methods  studied in the team  for dealing  with complex  geometries and heterogeneous propagation  media.  Moreover,  discontinuous Galerkin  discretization methods  are  naturally  adapted  to  local,  conforming  as  well  as non-conforming,  refinement  of  the  underlying  mesh,  Most  of  the existing DGTD  methods rely on  explicit time integration  schemes and lead to block diagonal mass matrices  which is often recognized as one of  the main  advantages  with regards  to  continuous finite  element methods.  However,  explicit DGTD  methods are  also constrained  by a stability condition  that can  be very  restrictive on  highly refined meshes  and  when  the  local  approximation  relies  on  high  order polynomial  interpolation.