The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressiblefluid exhibiting phase transitions between a liquid and a vapor phase in the presence ofcapillarity effects close to phase boundaries. Standard numerical discretizations areknown to violate discrete versions of inherent energy inequalities, thus leading tospurious dynamics of computed solutions close to static equilibria (e.g.,parasitic currents). In this work, we propose a time-implicit discretization of theproblem, and use piecewise linear (or bilinear), globally continuous finite element spacesfor both, velocity and density fields, and two regularizing terms where correspondingparameters tend to zero as the mesh-size h > 0 tends to zero.Solvability, non-negativity of computed densities, as well as conservation of mass, and adiscrete energy law to control dynamics are shown. Computational experiments are providedto study interesting regimes of coefficients for viscosity and capillarity.

For robust discretizations of the Navier-Stokes equations with small viscosity, standardGalerkin schemes have to be augmented by stabilization terms due to the indefiniteconvective terms and due to a possible lost of a discrete inf-sup condition. For optimalcontrol problems for fluids such stabilization have in general an undesired effect in thesense that optimization and discretization do not commute. This is the case for thecombination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilizedPetrov-Galerkin (PSPG). In this work we study the effect of different stabilized finiteelement methods to distributed control problems governed by singular perturbed Oseenequations. In particular, we address the question whether a possible commutation error inoptimal control problems lead to a decline of convergence order. Therefore, we givea priori estimates for SUPG/PSPG. In a numerical study for a flow withboundary layers, we illustrate to which extend the commutation error affects theaccuracy.

It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instancestreamline upwind Petrov-Galerkin (SUPG), are optimal on isotropicmeshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. Wedescribe the new methodand prove an a priori error estimate. This method leads on anisotropic meshes to qualitatively better convergence behavior than other isotropicstabilization methods.The capability of the methodis illustrated by means of two numerical test problems.

In the second part of the paper, we compare the solutions producedin the framework of the conference “Mathematical and numericalaspects of low Mach number flows” organized by INRIA and MAB inPorquerolles, June 2004, to the reference solutions described inPart 1. We make some recommendations on how to produce goodquality solutions, and list a number of pitfalls to be avoided.

There are very few reference solutions in the literature onnon-Boussinesq natural convection flows. We propose here a testcase problem which extends the well-known De Vahl Davisdifferentially heated square cavity problem to the case of largetemperature differences for which the Boussinesq approximation isno longer valid. The paper is split in two parts: in this firstpart, we propose as yet unpublished reference solutions for casescharacterized by a non-dimensional temperature difference of 0.6,Ra = 106 (constant property and variable property cases) andRa = 107 (variable property case). These reference solutions wereproduced after a first international workshop organized by CEA andLIMSI in January 2000, in which the above authors volunteered toproduce accurate numerical solutions from which the presentreference solutions could be established.