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This paper is a set of lecture notes for a short introductory course on homogenization.
It covers the basic tools of periodic homogenization (two-scale asymptotic expansions, the
oscillating test function method and two-scale convergence) and briefly describes the main
results of the more general theory of G− or
H−convergence. Several applications of the method are given: derivation
of Darcy’s law for flows in porous media, derivation of the porosity...
We propose a mixed formulation for non-isothermal Oldroyd–Stokes problem where the both
extra stress and the heat flux’s vector are considered. Based on such a formulation, a
dual mixed finite element is constructed and analyzed. This finite element method enables
us to obtain precise approximations of the dual variable which are, for the non-isothermal
fluid flow problems, the viscous and polymeric components of the extra-stress tensor, as
well...
This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound...
We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.
We analyze a new formulation of the Stokes equations in
three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to
the angular variable: the problem for each Fourier coefficient is two-dimensional and has
six scalar unknowns, corresponding to the vector potential and the vorticity. A
spectral discretization is built on this formulation, which leads to an exactly
divergence-free discrete velocity. We prove optimal error estimates.
This study is mainly dedicated to the development and analysis of non-overlapping domain decomposition methods for solving continuous-pressure finite element formulations of the Stokes problem. These methods have the following special features. By keeping the equations and unknowns unchanged at the cross points, that is, points shared by more than two subdomains, one can interpret them as iterative solvers of the actual discrete problem directly issued from the finite element scheme. In this way,...
This study is mainly dedicated to the development and analysis of
non-overlapping domain decomposition methods for solving continuous-pressure
finite element formulations of the Stokes problem. These methods have the
following special features. By keeping the equations and unknowns unchanged at
the cross points, that is, points shared by more than two subdomains, one can
interpret them as iterative solvers of the actual discrete problem directly
issued from the finite element scheme. In this way,...
The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.
The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.
We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the...
The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.
The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.
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