High-order discontinuous Galerkin methods on simplicial meshes

- High-order polynomial basis functions on simplicial meshes

*p*methods on unstructured triangular (2D case) and tetrahedral (3D case) meshes where the degree

*p*of polynomials varies locally in space. This strategy is justified by the fact that a high order approximation is rarely needed everywhere in the computational domain particularly when the physical domain is unbounded or if singularities are present in the solution. Thus using a

*p*-adaptive strategy will result in a more efficient method.

- DG methods on non-conforming simplicial meshes

- Numerical treatment of complex materials with DG methods

- DG methods for time-harmonic wave propagation

Hybrid explicit/implicit time integration schemes

Explicit time integration schemes are subjected to stability conditions that become very restrictive when the underlying mesh is locally refined since the global time step is deduced from the volume of the smallest mesh element. Two main strategies can be considered to improve this situation: local time stepping and implicit time integration. Although the adoption of an implicit time integration scheme will allow to overcome the restrictive constraint on the time step for locally refined meshes, it is still not clear whether the resulting numerical methodology will demonstrate a superiority in terms of accuracy and overall computing cost over the original methodology based on an explicit time integration scheme. On one hand, the dispersion error should be minimized while taking care to the increase in complexity of the time integration technique. On the other hand, the computing cost is also directly impacted by the fact that at each time step, an implicit time integration scheme yields the inversion of a large sparse linear system. For certain linear systems of partial differential equations, the matrix of this system is constant as far as the time step is fixed during the simulation. The linear system solver can certainly exploit this fact but this will probably not translate into a drastic reduction of the cost of a single time step. Taking into account all these issues, a locally implicit time integration scheme could be the best compromise when solving an unsteady wave propagation problem on a locally refined mesh. The resulting hybrid explicit/implicit time integration strategy raises several challenges both from the mathematical analysis viewpoint (stability and accuracy, especially for what concern numerical dispersion) and from the computer implementation viewpoint (data structures, parallel computing aspects). Our activities in this domain aim at the design of hybrid explicit/implicit integration schemes in conjunction with high order discontinuous Galerkin methods on locally refined triangular or tetrahedral meshes.

Domain decomposition methods for wave propagation problems Our research activities on this topic aim at the formulation, analysis and concrete evaluation of Schwarz type domain decomposition methods in conjunction with discontinuous Galerkin approximation methods on unstructured simplicial meshes for the calculation of time-domain and time-harmonic wave propagation problems in heterogeneous media. Ongoing works in this direction are concerned with the design of overlapping and non-overlapping Schwarz algorithms for the solution of the time-harmonic Maxwell equations. In these algorithms, a first order absorbing condition is imposed at the interfaces between neighboring subdomains. This interface condition is equivalent to a Dirichlet condition for characteristic variables associated to incoming waves. For this reason, it is often referred as a natural interface condition. Whatsoever is the overlapping strategy, 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 iterative resolution of the interface system can be performed in a more efficient way using a Krylov method. Beside Schwarz algorithms based on natural interface conditions, we also study algorithms that make use of more effective transmission conditions. From the theoretical point of view, this represents a much more challenging goal since most of the existing results on optimized Schwarz algorithms have been obtained for scalar partial differential equations. We plan to extend the techniques for obtaining optimized Schwarz methods previously developed for the scalar partial differential equations to systems of partial differential equations by using appropriate relationships between systems and equivalent scalar problems.

High performance numerical computing

Beside basic research activities related to the topics discussed above, we are also committed to assessing the numerical methods and resolution algorithms that we propose through the numerical simulation of large-scale three-dimensional problems pertaining to computational electromagnetics and computation geoseismics. In practice, accuracy constraints bring the challenge of processing very large data sets corresponding to high resolution discretized geometrical models. In addition, the physical phenomena of interest are in essence unsteady and stability issues impose rather small time steps meaning that numerical simulations require a lot of time iterations. In this context, parallel computing is a mandatory path. Nowadays, modern parallel computing platforms most often take the form of clusters of multiprocessor systems which can be viewed as an hybrid distributed-shared memory systems. Moreover, multiple core systems are increasingly adopted thus introducing an additional level in the local memory hierarchy. Developing numerical algorithms that efficiently exploit such platforms raise several challenges, especially in the context of a massive parallelism. In this context, our activites are concered with (a) the exploitation of multiple levels of parallelism in domain decomposition algorithms for solving the large algebraic systems resulting from the DG discretization of the systems of PDEs that we consider and, (b) the study hierachical SPMD (Single Program Multiple Data) strategies for the parallelization of unstructured mesh based solvers.