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@PHDTHESIS{Voulis:761464,
author = {Voulis, Igor},
othercontributors = {Reusken, Arnold and Stevenson, Rob and Melcher, Christof},
title = {{A} space-time approach to two-phase stokes flow:
well-posedness and discretization},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
reportid = {RWTH-2019-04874},
pages = {1 Online-Ressource (136 Seiten)},
year = {2019},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University; Dissertation, RWTH Aachen University, 2019},
abstract = {In this thesis we consider a time-dependent Navier-Stokes
two-phase flow. A standard sharp interface model for the
fluid dynamics of two-phase flows is studied both from an
analytical and a numerical perspective. The Navier-Stokes
interface problem has discontinuous density and viscosity
coefficients. In such a setting the pressure solution and
gradient of the velocity solution are discontinuous across
an evolving interface. A closely related linear problem is
the two-phase Stokes problem. Despite the fact that this
linear Stokes interface problem is a strong simplification
of the two-phase Navier-Stokes flow, it is a good model
problem for the development of numerical methods. We are
particularly interested in a well-posed variational
formulation of this Stokes interface problem in a Eulerian
setting. We prefer a Eulerian formulation of the Stokes
interface problem because we discretize the problem in
Euclidean coordinates. Several well-posed formulations are
considered. We prove the well-posedness of a variational
space-time formulation in suitable spaces of divergence free
functions. A variant with a pressure Lagrange multiplier is
also considered. With a discontinuous Galerkin (DG) method
in mind, we formulate a well-posed discontinuous-in-time
version of the problem. The discontinuous-in-time
variational formulation involving the pressure variable for
the divergence free constraint is a natural starting point
for a space-time finite element discretization. Such methods
are discussed in an abstract setting in this thesis. We
consider discontinuous Galerkin time discretization methods
for abstract parabolic problems with inhomogeneous linear
constraints. This includes the Stokes problem with an
inhomogeneous (time-dependent) Dirichlet boundary condition
and/or an inhomogeneous divergence constraint. Another
problem of this kind is the heat equation with an
inhomogeneous boundary condition. Two common ways of
treating abstract saddle-point problems exist, namely
explicit or implicit (via Lagrange multipliers). Therefore,
different variational formulations of the parabolic problem
with constraints are introduced. For these formulations,
different modifications of a standard discontinuous Galerkin
time discretization method are considered. Different ways of
treating the linear constraints, e.g. ~by using an
appropriate projection, are introduced and analyzed. For
these discretizations, optimal error bounds, including
superconvergence results, are derived. Discretization error
bounds for the Lagrange multiplier are presented. Results of
experiments confirm the theoretically predicted optimal
convergence rates and show that without a modification the
(standard) DG method has suboptimal convergence behavior. We
consider two explicit examples: the heat equation and the
(two-phase) Stokes problem. Fully discrete schemes are
discussed in both cases, where the temporal DG scheme is
combined with a spatial continuous Galerkin (CG) scheme. For
the heat equation we show an optimal error bound with
respect to the energy norm. For the Stokes problem a dynamic
spatial mesh is considered because it is a useful tool to
limit the computational cost for two-phase flow problems
where a fine mesh is only necessary near the moving
interface. In the case of the one-phase Stokes problem, we
show global error bounds which are locally optimal. This is
done for the velocity and for the pressure Lagrange
multiplier. A space-time scheme for the two-phase Stokes
problem is introduced, including a discrete temporal
derivative with a discontinuous time-dependent coefficient.
Several numerical experiments are performed in the software
package DROPS. Standard finite element spaces have a poor
approximation quality for discontinuous unknowns. We show
the merit of the use of an extended finite element method.
This allows us to treat the discontinuity in pressure and
gives us an improved method.},
cin = {111710 / 110000},
ddc = {510},
cid = {$I:(DE-82)111710_20140620$ / $I:(DE-82)110000_20140620$},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2019-04874},
url = {https://publications.rwth-aachen.de/record/761464},
}
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