We introduce a finite volume scheme for the two-dimensional incompressible
Navier-Stokes equations. We use a triangular mesh. The unknowns for the
velocity and pressure are respectively piecewise constant and affine. We use a
projection method to deal with the incompressibility constraint. We show that
the differential operators in the Navier-Stokes equations and their discrete
counterparts share similar properties. In particular we state an inf-sup
(Babuska-Brezzi) condition. Using these properties we infer the stability of
the scheme.

We consider a finite volume scheme for the two-dimensional incompressible
Navier-Stokes equations. We use a triangular mesh. The unknowns for the
velocity and pressure are respectively piecewise constant and affine. We use a
projection method to deal with the incompressibility constraint. In a former
paper, the stability of the scheme has been proven. We infer from it its
convergence.

We introduce a finite volume scheme for the two-dimensional incompressible
Navier-Stokes equations. We use a triangular mesh. The unknowns for the
velocity and pressure are both piecewise constant (colocated scheme). We use a
projection (fractional-step) method to deal with the incompressibility
constraint. We prove that the differential operators in the Navier-Stokes
equations and their discrete counterparts share similar properties. In
particular, we state an inf-sup (Babuska-Brezzi) condition. We infer from it
the stability of the scheme.

We present a Lohner type algorithm for the computation of rigorous bounds for
solutions of ordinary differential equations and its derivatives with respect
to initial conditions up to arbitrary order. As an application we prove the
existence of multiple invariant tori around some elliptic periodic orbits for
the pendulum equation with periodic forcing and for Michelson system.

We show that the sharp constant in the classical $n$-dimensional Hardy-Leray
inequality can be improved for axisymmetric divergence-free fields, and find
its optimal value. The same result is obtained for $n=2$ without the
axisymmetry assumption.

Every finite branch solutions to the sixth Painleve equation around a fixed
singular point is an algebraic branch solution. In particular a global solution
is an algebraic solution if and only if it is finitely many-valued globally.
The proof of this result relies on algebraic geometry of Painleve VI,
Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces,
resolutions of Kleinian singularities, and power geometry of algebraic
differential equations. In the course of the proof we are also able to classify
all finite branch solutions up to Backlund transformations.