The two subjects of my thesis are firstly the study of the asymptotic behavior of some singular perturbation problems in a general (curved) bounded domain, and secondly, the stability and convergence of the cell-centered Finite Volume (FV) method using the Taylor Series Expansion Scheme (TSES) without fictitious domains.

For the singular perturbation problems, we first consider the reaction-diffusion and heat equations, in a general bounded domain Ω of [special characters omitted], d ≥ 2, with zero Dirichlet boundary condition and a small diffusion coefficient ϵ in front of the Laplacian. Using curvilinear coordinates adapted to the boundary of Ω, for both the reaction-diffusion and heat problems, we obtain an asymptotic expansion with respect to ϵ at all orders. In the end, we see that the convergence results, between the exact and approximate solutions, are optimal. We also study, in a general bounded domain in [special characters omitted] with a characteristic boundary, the asymptotic behavior of the linearized Navier-Stokes equations (LNSE) when the viscosity is small. Using curvilinear coordinates again, we show that the solutions of the LNSE behave like the corresponding Euler solutions except in a thin region, near the boundary, where a certain heat solution is added as a corrector.

Concerning the second topic, it is well known that the stability of the FV method is easy to obtain, but proving its convergence is problematic because the FV method is only weakly consistent (in the sense of weak vs strong convergence). Some authors have proven the convergence by using compactness arguments, even for linear problems. Instead, we prove here the convergence by comparing the FV method with the Finite Difference (FD) method based on the same mesh. The results have been obtained for a rectangular domain (0, 1)² in [special characters omitted] with rectangular meshes.