Discretization Methods: Multiscale and Stabilized Methods


Volker Gravemeier, Ursula Rasthofer and Benedikt Schott

This method is particularly supported by the Emmy Noether Research Group.


In the 1970s and 1980s, stabilized finite element methods were proposed for flow problems. Those methods were developed in response to two major problems which had been observed when using standard Galerkin finite element methods in fluid mechanics, that is, problems in the case of dominating convection and the required fulfillment of a so-called inf-sup condition for (incompressible) flow problems. Later, in the 1990s, a more comprehensive framework in the form of the variational multiscale method was developed. Stabilized finite element methods could then be derived from the variational multiscale framework. The aforementioned problems may all be related to unresolved scales, that is, to scales which the respectively chosen discretizations are not able to represent. Using variational multiscale methods, those unresolved scales, or at least their effect, may be brought into the problem formulation to be solved based on the chosen discretization. This is achieved by a consistent separation of the overall function spaces into resolved and unresolved subspaces via variational projection, and without resorting to any ad hoc measures. In general, the variational multiscale method may be considered as a method particularly suited for problems with broad scale ranges. Variants of the variational multiscale method have been particularly generated for the challenging problem of turbulent flow. Recently, we developed the Algebraic Variational Multiscale-Multigrid Method (AVM3) for turbulent flow simulation. However, the variational multiscale has also been used as a basis for developing computational methods for various other applications beyond flow problems in the meantime. Multiscale and stabilized methods are used in the following research fields, methods and applications, respectively



Please find publications on this topic here.