Section: Scientific Foundations

Numerical tools: High-order discretization methods

The discontinuous Galerkin finite element method (DGFEM) offers interesting perspectives, since it gives a framework for the combination of techniques developed in the incompressible finite-element (well-founded treatment of incompressibility constraints, pressure approximation, and stabilization for high-Reynolds-number flows) and the compressible finite-volume community (entropy solutions, Riemann solvers and flux limiters).

In addition, the order limit of finite volume discretizations is broken by the variational formulation underlying DGFEM, which makes it possible to develop discretization schemes with local mesh refinement and local variation of the polynomial degree (hp -methods). At the same time, the well-established finite-element knowledge for saddle-point problems can be set on work.

Noting that different approaches based on discontinuous Galerkin methods have been used in recent years for the solution of challenging flow problems. Since the project team members have experience with these and other stabilized finite element methods, a combination of the different techniques is expected to be beneficial in order to gain efficiency.

It is generally accepted that an important advantage of DGFEM beside its flexibility is the fact that it is locally conservative. At the same time, its drawback is its relatively high numerical cost. For example, compared to continuous P¹ finite elements on a triangular mesh, the number of unknowns are increased by a factor of 6 (and a factor of 2 with respect to the Crouzeix-Raviart space); considering the system matrix even leads to a more disadvantageous count. Concerning higher-order spaces, standard DGFEM has a negligible overhead for polynomial orders starting from p = 5 , which is probably not the most employed in practice. The question of how to increase efficiency of DGFEM is an important topic of recent research. Our approach in this field is based on comparison with stabilized FE methods.

There are many similarities between DGFEM and SDFEM (streamline-diffusion FEM) based on piecewise linear elements for the transport equation from a theoretical point of view, see for example the classical text book [57] . Recently, attempts have been made to shed brighter light on the relations between these methods [49] [50] , [51] . A better understanding of the relations between these methods will contribute to the development of more efficient schemes with desired properties. As outlined before, the goal is to cut down the computational overhead of standard DGFEM, while retaining its robustness and conservation properties.

Formulation of discretization schemes based on discontinuous finite element spaces is nowadays standard. However, some important questions remain to be solved:

• How to combine possibly higher-order spaces with special numerical integration in order to obtain fast computation of residuals and matrices ? How to stabilize such higher-order DGFEM ?

• Treatment of quadrilateral and hexahedral meshes: Hexahedral meshes are economical for simple geometries. However, arbitrary hexahedra (the image of the unit cube under a trilinear transformation) lead to challenging questions of discretization. For example, straightforward generalization of some standard methods such as mixed finite elements may lead to bad convergence behavior [39] .

• Time-discretization: In order to be fully conservative, the time discretization has to be implicit: for example for the transport equation it seems reasonable not to distinguish between time and space variables and it is therefore natural to discretize both time and space with discontinuous finite elements. The choice of DG time-discretization is natural in view of its good stability and conservation properties. However, the higher-order members of this family lead to coupled systems which have to be solved in each time-step.

• Solution of the discrete systems: The computing time largely depends on the way the discrete nonlinear and linear systems are solved. Concerning the solution of the nonlinear systems, we observe that the systems arising in our applications require special solvers, using homotopy methods, time-stepping and specially tuned Newton algorithms. In each step of the algorithm, the solution of the linear systems is a major bottleneck for adaptive high-order method. In order to gain efficiency, the hierarchical structure of the discretization should be exploited, which requires a close connection between numerical schemes and linear solvers.