High-order discontinuous Galerkin methods on simplicial meshes

  • High-order polynomial basis functions on simplicial meshes
We are interested in exploiting hierarchical polynomials since we believe that they are well suited to the design of high-order DG-Pp 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
A mesh (made of triangles for example) is said to be conforming if the intersection of two triangles (tetrahedra in 3D) is either empty, a vertex or a whole edge (face in 3D). Three main reasons motivate our investigation of DG methods able to deal with non-conforming simplicial meshes: (1) the final mesh is made of separately constructed meshes and a conforming assembling is a very hard task, (2) the presence of geometrical features (fine structures, corners, etc.) or physical ones (multi-scale problems, etc.) requiring a highly localized refinement and, (3) the construction of hybrid meshes mixing elements of different types such as hexahedra and tetrahedra without the addition of another type of element (a pyramid for example in this case).
  • Numerical treatment of complex materials with DG methods
Realistic applications of electromagnetic wave propagation and seismic wave propagation involve complex materials (propagation media) characterized by parameters (e.g electrical permittivity or Lamé coefficients) which are functions of both space and time. Dispersive electromagnetic materials and viscoelastic geological materials are classical examples. These materials are modeled by differential equations wich must be added to the standard mathematical models of electromagnetic and sesimic wave propagation. We investigate the design of DG methods adapted to the resulting systems of partial differential equations with two main application contexts in mind: the propagation of electromagnetic waves in biological tissues and the propagation of sesimic waves in viscoelatic geological media.
  • DG methods for time-harmonic wave propagation
There exists several practical situations for which considering the formulation of a wave propagation problem in the frequency domain is highly justified if not the best option. This is for instance the case with electromagnetic waves propagation for furtivity applications or, when modeling seismic wave propagation for the imaging of complex structures. We investigate the design of DG methods for the discretization of the time-harmonic formulation of the Maxwell equations and elastodynamic equations in their first-order form.

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.