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The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressible
fluid exhibiting phase transitions between a liquid and a vapor phase in the presence of
capillarity effects close to phase boundaries. Standard numerical discretizations are
known to violate discrete versions of inherent energy inequalities, thus leading to
spurious dynamics of computed solutions close to static equilibria (e.g.,
parasitic currents). In this work, we propose a time-implicit discretization of the
problem, and use piecewise linear (or bilinear), globally continuous finite element spaces
for both, velocity and density fields, and two regularizing terms where corresponding
parameters 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 a
discrete energy law to control dynamics are shown. Computational experiments are provided
to study interesting regimes of coefficients for viscosity and capillarity.
For robust discretizations of the Navier-Stokes equations with small viscosity, standard
Galerkin schemes have to be augmented by stabilization terms due to the indefinite
convective terms and due to a possible lost of a discrete inf-sup condition. For optimal
control problems for fluids such stabilization have in general an undesired effect in the
sense that optimization and discretization do not commute. This is the case for the
combination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilized
Petrov-Galerkin (PSPG). In this work we study the effect of different stabilized finite
element methods to distributed control problems governed by singular perturbed Oseen
equations. In particular, we address the question whether a possible commutation error in
optimal control problems lead to a decline of convergence order. Therefore, we give
a priori estimates for SUPG/PSPG. In a numerical study for a flow with
boundary layers, we illustrate to which extend the commutation error affects the
It is well known that the classical local projection
method as well as residual-based stabilization techniques, as for instance
streamline upwind Petrov-Galerkin (SUPG), are optimal on isotropic
meshes. Here we extend the local projection stabilization for the Navier-Stokes
system to anisotropic quadrilateral meshes in two spatial dimensions. We
describe the new method
and prove an a priori error estimate.
This method leads on anisotropic meshes to qualitatively better
convergence behavior than other isotropic
The capability of the method
is illustrated by means of two numerical test problems.
In the second part of the paper, we compare the solutions produced
in the framework of the conference “Mathematical and numerical
aspects of low Mach number flows” organized by INRIA and MAB in
Porquerolles, June 2004, to the reference solutions described in
Part 1. We make some recommendations on how to produce good
quality solutions, and list a number of pitfalls to be avoided.
There are very few reference solutions in the literature on
non-Boussinesq natural convection flows. We propose here a test
case problem which extends the well-known De Vahl Davis
differentially heated square cavity problem to the case of large
temperature differences for which the Boussinesq approximation is
no longer valid. The paper is split in two parts: in this first
part, we propose as yet unpublished reference solutions for cases
characterized by a non-dimensional temperature difference of 0.6,
Ra = 106 (constant property and variable property cases) and
Ra = 107 (variable property case). These reference solutions were
produced after a first international workshop organized by CEA and
LIMSI in January 2000, in which the above authors volunteered to
produce accurate numerical solutions from which the present
reference solutions could be established.
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