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Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry
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Section: New Results

Time integration strategies and resolution algorithms

Hybrid explicit/implicit DGTD-Im1 $\#8473 _p$ 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-Im1 $\#8473 _p$ 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-Im1 $\#8473 _p$ 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-Im3 $\#8473 _1$ 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 Im4 $\#8776 $ 0.2% of the tetrahedra of the mesh), the simulation time is reduced from Im4 $\#8776 $ 25 h to Im4 $\#8776 $ 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.

Figure 2. Scattering of a plane wave by a Falcon jet geometry. Contour lines of Ez and Im5 ${|\#119812 |}$ on the aircraft surface

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-Im1 $\#8473 _p$ 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-Im1 $\#8473 _p$ 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-Im1 $\#8473 _p$ 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-Im1 $\#8473 _p$ method.


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