NONSTANDARD DISCRETIZATIONS FOR FLUID FLOWS

Computational Fluid Dynamics is a field on the interface between Numerical Analysis, Computational Science, Physics and Engineering. It underwent an intensive development primarily because of the needs of aeronautics, nuclear physics and engineering, geophysics, chemical and automotive engineering etc., and the rapid development of fast computers. The major efforts are concentrated on the efficient numerical approximation of the solution of the incompressible, quasi-compressible and compressible Navier-Stokes or Euler equations, sometimes in conjunction with additional advection-diffusion systems for transport of heat or substances. Relatively recently, the scope of problems was widened by considering fluid structure interaction, MHD and non-Newtonian fluids, which introduced some major new challenges. In addition, the development of models for multiphase flows, biofluids, micro/nanofluids is a challenge by itself, and it is far from being completed yet. The focus of this workshop was on the development and analysis of new algorithms as well as the practical solution of some very challenging physical problems (which themselves require some non-standard thinking even if more classical techniques are used). The common ground for all these models is that they comprise nonlinear advection-diffusion(-reaction) equations with linear constraints. Depending on the parameters of the system, they can exhibit the whole scale from a predominantly parabolic/elliptic behaviour (low Reynolds number flows) up to a predominantly hyperbolic behaviour (the Euler system). Other challenges are the (sometimes very strong) non-linearity of the problems, the presence of constraints which necessarily lead to saddle point systems, the appearance of more than one spatial and/or temporal scale, etc. All these make the development of a universal and optimal algorithm impossible and stimulated the development of various methods for the various classes of problems. Several major classical discretization techniques have been developed, based on finite difference, finite element, finite volume and spectral methods. More recently, we witnessed the development of some new interesting techniques like the hp finite elements, discontinuous Galerkin methods and other non-conforming or div-conformingmethods, unstructured finite volumes, domain decomposition and mortar techniques. In case of advection-dominated flows, these techniques are combined with various stabilization mechanisms: edge stabilization; residualand entropy-based viscosity; and special fluxes for discontinuous Galerkin schemes. These are closely related to some recent approaches for turbulence modelling like subgrid viscosity, variational multiscale methods and large eddy simulation. A major development in all these areas was the idea for a posteriori error estimation and nonlinear adaptivity. The discretization of time dependent problems