Finite Element Approximations For Fluid Flows Governed By Nonlinear Slip Boundary Conditions Of Friction Type
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Finite Element Approximations for Fluid Flows Governed by Nonlinear Slip Boundary Conditions of Friction Type
This thesis is divided in three main chapters devoted to the study of finite element approximations of fluid flows with special nonlinearities coming from boundary con- ditions. In Chapter 1, we consider the finite element approximations of steady Navier-Stokes and Stokes equations driven by threshold slip boundary conditions. After re-writing the problems in the form of variational inequalities, a fixed point strategy is used to show existence of solutions. Next we prove that the finite element approximations for the Stokes and Navier Stokes equations converge respectively to the solutions of each continuous problem. Finally, Uzawa’s algorithm is formulated and convergence of the procedure is shown, and numerical validation tests are achieved. Chapter 2 is concerned with the finite element approximation for the stationary power law Stokes equations driven by slip boundary conditions of “friction type”. It is shown that by applying a variant of Babuska-Brezzi’s theory for mixed problems, convergence of the finite element approximation formulated is achieved with classi- cal assumptions on the regularity of the weak solution. Solution algorithm for the mixed variational problem is presented and analyzed in details. Finally, numerical simulations that validate the theoretical findings are exhibited. In Chapter 3, we are dealing with the study of the stability for all positive time of Crank-Nicolson scheme for the two-dimensional Navier-Stokes equation driven by slip boundary conditions of “friction type”. We discretize these equations in time using the Crank-Nicolson scheme and in space using finite element approximation. We prove that the numerical scheme is stable in L2 and H1-norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition.
Deterministic and Stochastic Optimal Control and Inverse Problems
Inverse problems of identifying parameters and initial/boundary conditions in deterministic and stochastic partial differential equations constitute a vibrant and emerging research area that has found numerous applications. A related problem of paramount importance is the optimal control problem for stochastic differential equations. This edited volume comprises invited contributions from world-renowned researchers in the subject of control and inverse problems. There are several contributions on optimal control and inverse problems covering different aspects of the theory, numerical methods, and applications. Besides a unified presentation of the most recent and relevant developments, this volume also presents some survey articles to make the material self-contained. To maintain the highest level of scientific quality, all manuscripts have been thoroughly reviewed.
Mathematics of Large Eddy Simulation of Turbulent Flows
Author: Luigi Carlo Berselli
language: en
Publisher: Springer Science & Business Media
Release Date: 2006
The LES-method is rapidly developing in many practical applications in engineering The mathematical background is presented here for the first time in book form by one of the leaders in the field