Finite element approximations for fluid flows governed by nonlinear slip boundary conditions of friction type: from theory to computations

Abstract

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.