Some Remarks on Residual-based Stabilisation of Inf-sup Stable Discretisations of the Generalised Oseen Problem

Abstract

We consider residual-based stabilised finite element methods for the generalised Oseen problem. The unique solvability based on a modified stability condition and an error estimate are proved for inf-sup stable discretisations of velocity and pressure. The analysis highlights the role of an additional stabilisation of the incompressibility constraint. It turns out that the stabilisation terms of the streamline-diffusion (SUPG) type play a less important role. In particular, there exists a conditional stability problem of the SUPG stabilisation if two relevant problem parameters tend to zero. The analysis extends a recent result to general shape-regular meshes and to discontinuous pressure interpolation. Some numerical observations support the theoretical results.

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

“…The SUPG method was introduced in [81,28] for stabilizing scalar convectiondominated convection-diffusion equations. Stabilizations of the Oseen equations and the stationary Navier-Stokes equations which contain the SUPG term were analyzed in [72,111,136,53], and extensions of the analysis can be found in [137,110,114]. Surveys of the results are provided in [22,124].…”

Enabling numerical accuracy of Navier-Stokes-αthrough deconvolution and enhanced stability

“…In practice this term is often omitted, and until recently it was not clear if it is needed for technical reasons of the analysis or played an important role in computations. The role of the grad-div stabilization was again emphasized in the recent studies of the (stabilized) finite element methods for incompressible flow problems, see [21,38,43,50], also in conjunction with the rotation form of nonlinearities in the Navier-Stokes equations [33,34,42] and variational multiscale turbulence modelling [27]. Its relation to the variational multiscale approach is revealed in [15,22,44].…”

Discrete and continuous hamiltonian systems for wave modelling

“…Recombining, the various terms yields 71) which after cancellation of terms with opposite sign gives the same result as (4.69)…”

“…A direct DGFEM discretisation of the incompressible Navier-Stokes equations (or the Euler equations as special case) generally requires the inf-sup condition to be satisfied to attain numerical stability [39,71]. In order to get a stable pressure approximation, two different strategies are often pursued: either a pressure stabilisation term is used or the approximation spaces for velocity and pressure are chosen (differently) such that an inf-sup compatibility condition is fulfilled.…”