Computer and Computational Sciences at Exascale

The discontinuous Galerkin (DG) method was introduced in 1973 by Reed and Hill to solve the neutron transport equation. From this time to the 90’s a survey of the DG methods would likely fit into one page. In the meantime, the finite volume approach has been widely adopted by computational fluid dynamics scientists and has now nearly supplanted classical finite difference and finite element methods in solving problems of non-linear convection. The success of the finite volume method is due to its ability to capture discontinuous solutions which may occur when solving non-linear equations or more simply, when convecting discontinuous initial data in the linear case. Let us first remark that DG methods share with finite volumes this property since a first order finite volume scheme can be viewed as a 0th order DG scheme. However a DG method may be also considered as a finite element one where the continuity constraint at an element interface is released. While it keeps almost all the advantages of the finite element method (large spectrum of applications, complex geometries, etc.), the DG method has other nice properties which explain the renewed interest it gains in various domains in scientific computing nowadays. In short, it is a compact support, high order finite element method (it relies on a high order interpolation of the problem unknowns within each cell of the mesh) that, from the algorithmic point of view, combines sparse and dense linear algebra operations. In particular, the DG method is naturall