From the Navier-Stokes/Brinkman model, a penalization method has been derived by several authors to compute incompressible Navier-Stokes equations around obstacles. In this paper, convergence theorems and error estimates are derived for two kinds of penalization. The first one corresponds to a L 2 penalization inducing a Darcy equation in the solid body, the second one corresponds to a H 1 penalization and induces a Brinkman equation in the body. Numerical tests are performed to confirm the efficiency and accuracy of the method.
Preface v Contents ix I The equations of fluid mechanics 1 Continuous description of a fluid 1.1 The continuous medium assumption. Density 1.2 Lagrangian and Eulerian coordinates 2 The transport theorem 3 Evolution equations 3.1 Balance equations 3.2 Cauchy's stress theorem 3.3 Evolution equations revisited 4 Fundamental laws: Newtonian fluids and .thermodynamics laws 4.1 Fluids at rest 4.2 Newton's hypothesis 4.3 Consequences of the second law of thermodynamics 4.4 Equation for the specific internal energy 4.5 Formulation in entropy and temperature 5 Summary of the equations 6 Incompressible models 6.1 The incompressibility assumption 6.2 Overview of the incompressible models 7 Some exact steady solutions 7.1 Poiseuille flow in a pipe 7.2 Planar shear flow 7.3 Couette flow between two cylinders II Analysis tools 1 Main notation 2 Fundamental results from functional analysis 51 ix
Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.
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