Section: New Results
Time integration strategies and resolution algorithms
Hybrid explicit/implicit DGTD-
method
Participants : Stéphane Descombes, Victorita Dolean, Stéphane Lanteri, Jan Verwer [ Department Modelling, Analysis and Simulation, CWI, The Netherlands ] .
Existing numerical methods for the solution of the time domain Maxwell
equations often rely on explicit time integration schemes and are
therefore constrained by a stability condition that can be very
restrictive on highly refined meshes. An implicit time integration
scheme is a natural way to obtain a time domain method which is
unconditionally stable[3] . Starting from the
explicit, non-dissipative, DGTD- method introduced in
[8] , we have proposed to use of Crank-Nicholson
scheme in place of the explicit leap-frog scheme adopted in this
method. As a result, we obtain an unconditionally stable,
non-dissipative, implicit DGTD- method, but at the
expense of the inversion of a global linear system at each time step,
thus obliterating one of the attractive features of discontinuous
Galerkin formulations. A more viable approach for 3D simulations
consists in applying an implicit time integration scheme locally i.e
in the refined regions of the mesh, while preserving an explicit time
scheme in the complementary part, resulting in an hybrid
explicit-implicit (or locally implicit) time integration strategy. As
a preliminary step in this direction [21] , we have
studied a hybrid explicit-implicit DGTD method initially introduced by
Piperno in [41] . An illustration of the application
of the resulting hybrid explicit-implicit DGTD-
method
is shown on Fig. 2 below. The underlying
tetrahedral mesh consists of 360,495 vertices and 2,024,924 elements.
When 6381 elements are treated implicitly (i.e 
0.2% of the
tetrahedra of the mesh), the simulation time is reduced from
25 h to 4 h. Besides, such a hybrid,
Crank-Nicholson/Leap-Frog time integration scheme is well known to the
ODE community and has indeed recently been studied in the context of
component splitting methods by J. Verwer at CWI [47] .
Although similar in the building ingredients, the schemes in
[41] and [47] exhibit differences
which motivate further investigations. We have initiated this year a
collaboration in this direction with J. Vewer with whom we also plan
to study the possibility of designing higher order hybrid
explicit-implicit time schemes.
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Optimized Schwarz algorithms for the frequency domain Maxwell equations
Participants : Victorita Dolean, Mohamed El Bouajaji, Martin Gander [ Mathematics Section, University of Geneva ] , Stéphane Lanteri, Ronan Perrussel [ Ampère Laboratory, Ecole Centrale de Lyon ] .
The linear systems resulting from the discretization of the 3D time harmonic Maxwell equations using discontinuous Galerkin methods on simplicial meshes are characterized by large sparse, complex coefficients and irregularly structured matrices. Classical preconditioned iterative methods (such as the GMRES Krylov method preconditioned by an incomplete LU factorization) generally behave poorly on these linear systems. A standard alternative solution strategy calls for sparse direct solvers. However, this approach is not feasible for reasonably large systems due to the memory requirements of direct solvers. On the other hand, parallel computing is recognized as a mandatory path for the design of algorithms capable of solving problems of realistic importance. Several parallel sparse direct solvers have been developed in the recent years such as MUMPS [36] . Even if these solvers efficiently exploit distributed memory parallel computing platforms and allow to treat very large problems, there is still room for improvements of the situation. Iterative methods can be used to overcome this memory problem. The main difficulty encountered by these methods is their lack of robustness and, generally, the unpredictability and inconsistency of their performance when they are used over a wide range of different problems. Because an iterative solver will usually require fewer iterations and less time if more fill-in is allowed in the preconditioner, some approaches combine the direct solvers techniques with other iterative preconditioning techniques in order to build robust preconditioners. For example, a popular approach in the domain decomposition framework is to use a direct solver inside each subdomain and to use an iterative solver on the interfaces between subdomains.
Even if they have been introduced for the first time two centuries ago, over the last two decades, classical Schwarz methods have regained a lot of popularity with the developement of parallel computers. First developed for the elliptic problems, they have been recently extended to systems of hyperbolic partial differential equations, and it was observed that the classical Schwartz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. Over the last decade, optimized versions of Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains, and are also convergent without overlap for elliptic problems. The extension of such methods to systems of equations and more precisely to Maxwell's system (time harmonic and time discretized equations) has been done recently in [14] .
These new transmission conditions were originally proposed for three different reasons: first, to obtain Schwarz algorithms that are convergent without overlap; secondly, to obtain a convergent Schwarz method for the Helmholtz equation, where the classical Schwarz algorithm is not convergent, even with overlap; and third, to accelerate the convergence of classical Schwarz algorithms. Several studies towards the development of optimized Schwarz methods for the time harmonic Maxwell equations have been conducted this last decade, most often in combination with conforming edge element approximations. Optimized Schwarz algorithms can involve transmission conditions that are based on high order derivatives of the interface variables. However, the effectiveness of the new optimized interface conditions has been proved so far only for simple geometries and applications.
In order to extend them to more realistic applications and geometries,
and high order approximation methods, our first strategy for the
design of parallel solvers in conjunction with discontinuous Galerkin
methods on simplicial meshes relied on a Schwarz algorithm where a
classical condition is imposed at the interfaces between neighboring
subdomains which corresponds to a Dirichlet condition for
characteristic variables associated to incoming waves
[6] . From the discretization point of view,
this interface condition gives rise to a boundary integral term which
is treated using a flux splitting scheme similar to the one applied at
absorbing boundaries. The Schwarz algorithm can be used as a global
solver or it can be reformulated as a Richardson iterative method
acting on an interface system. In the latter case, the resolution of
the interface system can be performed in a more efficient way using a
Krylov method. Besides, results on the applications of optimized
Schwarz algorithms combined to a high order DGFD- method
on triangular meshes for the discretization of the 2D Maxwell
equations are reported in [7] .
The optimized interface conditions proposed in [14] were devised for the case of non-conducting propagation media. We have started this year a study aiming at the formulation of such conditions for conducting media [23] .
Algebraic preconditioning techniques for
a high order DGFD- method
Participants : Matthias Bollhoefer [ Institute of Computational Mathematics, TU Braunschweig ] , Luc Giraud [ HiePACS project-team, INRIA Bordeaux - Sud-Ouest ] , Stéphane Lanteri, Jean Roman [ HiePACS project-team, INRIA Bordeaux - Sud-Ouest ] .
For large 3D problems, the use of a sparse direct method for solving
the algebraic sparse system resulting from the discretization of the
frequency domain Maxwell equations by a high order DGFD-
method is simply not feasible because of the memory overhead, even if
these systems are associated to subdomain problems in a domain
decomposition setting. A possible alternative is to replace the
sparse direct method by a preconditioned iterative method for which an
appropriate preconditioning technique has to be designed. For this
purpose, we are investigating incomplete factorization methods that
exploit the block structure of the underlying matrices which is
directly related to the approximation order of the physical quantities
within each mesh element in the DGFD- method.
