In Chapter 1, we consider the 2.5 dimensional approximation (that is, with only three Galerkin finite elements in the y direction), to the inviscid primitive equations posed on a bounded domain. An infinite set of nonlocal boundary conditions is proposed, and with the use of the semi–group theory, the well–posedness of the linearized equations, when supplemented with this set of boundary conditions, is established.

In Chapter 2, we study the numerical simulation of the full (nonlinear) 2.5D inviscid primitive equations. An up–wind scheme is proposed. Simulations are performed on two nested domains. Non blow–up of the numerical results adds credentials to the conjecture that the nonlocal boundary conditions proposed in Chapter 1 are also suitable for the nonlinear equations. The fact that the numerical results match very well on the nested domains demonstrates the transparency property of the proposed boundary conditions.

In Chapter 3 we study the resolution of certain nonlinear diffusive partial differential equations, in the presence of incompatibilities between the initial and boundary conditions. We propose a set of correction procedures using suitable singular corner functions. The correction procedures allows to recover the full accuracy of the numerical schemes near the time-space corners, which would otherwise have been lost due to the presence of the incompatibilities there.