  A global time-discretized scheme for the Navier-Stokes equation system in its
Leray projection form is defined. It is shown that the scheme converges to a
bounded global classical solution for smooth data which have polynomial decay
at infinity. Furthermore, the algorithm proposed is extended to the situation
of initial-boundary value problems. Algorithms constructed in a different
context (cf. [4, 10, 5, 9]) may be used within the proposed scheme in order to
compute the solution of Leray's form of the Navier-Stokes system. The main idea
for global existence is to define a control function dynamically and show
explicitly that the scheme which solves a controlled Navier-Stokes type
equation can control the modulus of velocity and the first derivatives of
velocity to be bounded. The method described here can be extended to
Navier-Stokes equations on compact manifolds which is done in a subsequent
paper.
