Numerical Analysis Seminars: Stabilized finite element methods for high Reynolds flows: hydrodynamic stability and computability
18 November 2011 14:00 in E101
The computation of high Reynolds flows remains an important challenge in scientific computation. It is generally believed that so called Large Eddy Simulation, i.e. a simulation where only the large eddys are computed, is feasible and that useful information can be extracted from such a computation. The underlying idea is that the flow is dominated by large scale structures and that resolving those structures is sufficient for the computation of relevant quantities. Although there appears to be computational evidence giving support to this hypothesis, no analytical results, in the form of error estimates, validating the theory exist in the literature. One reason for this is the appearance of huge exponential constants reflecting the hydrodynamic instability. In this talk we will consider stabilized finite element methods as a tool for large eddy simulation. We will show that these methods allows for optimal error estimates that are independent of the viscosity for smooth solutions of the Navier-Stokes' equations. Unfortunately the constants of these estimates blow up in case the solution looses regularity, i.e. the solution develops layers or becomes turbulent. To get better insight in this instability, triggered by the nonlinearity, we will study the viscous Burgers' equation. First we show that the same huge constants appear for error estimates in the L2-norm. Then we consider L2-norm error estimates of filtered quantities, and show that in this case we can derive error estimates that have moderate constants, depending only on the initial data. This leads naturally to a relation between the filter width and the convergence order of the method.
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