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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no longer valid.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part, we propose as yet unpublished reference solutions for cases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...
Over the past decade or so, there have been a large number of modelling approaches aimed
at elucidating the most important mechanisms affecting the formation of new capillaries
from parent blood vessels — a process known as angiogenesis. Most studies have focussed
upon the way in which capillary sprouts are initiated and migrate in response to
diffusible chemical stimuli supplied by hypoxic stromal cells and leukocytes in the
contexts of solid tumour...
Bacillus subtilis swarms rapidly over the surface of a synthetic medium
creating remarkable hyperbranched dendritic communities. Models to reproduce such effects
have been proposed under the form of parabolic Partial Differential Equations representing
the dynamics of the active cells (both motile and multiplying), the passive cells
(non-motile and non-growing) and nutrient concentration. We test the numerical behavior of
such models and compare...
For scalar conservation laws in one space dimension with a flux function discontinuous in space, there exist infinitely many classes of solutions which are L1 contractive. Each class is characterized by a connection (A,B) which determines the interface entropy. For solutions corresponding to a connection (A,B), there exists convergent numerical schemes based on Godunov or Engquist−Osher schemes. The natural question is how to obtain schemes, corresponding to computationally less expensive monotone...
This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second...
We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the -gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the...
We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L2-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the...
In a recent paper [4] we have proposed and analysed a suitable mathematical model which describes the coupling of the Navier-Stokes with the Oseen equations. In this paper we propose a numerical solution of the coupled problem by subdomain splitting. After a preliminary analysis, we prove a convergence result for an iterative algorithm that alternates the solution of the Navier-Stokes problem to the one of the Oseen problem.
In a recent paper [4] we have proposed and analysed
a suitable mathematical model
which describes the coupling of the Navier-Stokes with the
Oseen equations.
In this paper we propose a numerical solution of the coupled
problem by subdomain splitting.
After a preliminary analysis, we prove a convergence result for
an iterative algorithm that alternates the solution of the Navier-Stokes
problem to the one of the Oseen problem.
We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale...
In this paper we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency-dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale ε and use two-scale homogenization techniques to derive effective...
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