Analysis and approximation of linear and nonlinear partial differential equations: Boundary layers, atmospheric equations, change of phase.

摘要

My thesis is in the area of mathematical analysis of computational fluid dynamics and geophysical fluid dynamics. My thesis contains three main objectives.;Theoretical and numerical analysis of the singularly perturbed problems are considered in Chapter 2 and 3. One major focus and contribution in this subject is to study the boundary layers of the convection-diffusion equations in the presence of characteristic points. Theoretical results on the boundary layers is used to improve on their computations. The numerical analysis of the singularly perturbed convection-diffusion equations raises substantial difficulties when we consider a standard finite element space. To remedy this difficulty, the profile of the boundary layer is introduced.;New avenues are explored to study the two dimensional inviscid primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions for the system of equations. Spatial discretization is done by first order finite volume methods. A version of the projection method is introduced to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. The resulting scheme allows for a significant reduction of the errors near the topography when compared to more standard finite volume schemes. In the numerical simulations, we first present the associated convergence results that are satisfied by the solutions simulated by our scheme when compared to particular analytic solutions. We then report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated.;It is well known that the solutions of the 3D Navier--Stokes equations remain bounded for all time if the initial data and the forcing are sufficiently small relative to the viscosity, and for a finite time given any bounded initial data. We consider two temporal discretisations (semi-implicit and fully implicit) of the 3D Navier--Stokes equations in a periodic domain and prove that their solutions remain bounded in H1 subject to essentially the same respective smallness conditions (on initial data and forcing or on the time of existence) as the continuous system and to suitable time-step restrictions.