On the Convergence of Discrete Approximations to the Navier-Stokes Equations (Classic Reprint)

I ntroduction. The Navler-S tokes equations, describing the motion of a viscous Incompressible fluid, can be written In the dlmenslonless form (1) Sv +grad p= -jji +Z +E, (v =3 j (2) dlv V= 0where the vector v, with components v., 1= 1,2,5, Is the velocity, p Isthe pressure, EI sthe external force, 3, denotes differentiation with respect to the timet and 5. denotes differentiation with respect to the space variable x., 1= 1,2,3. Vector quantities are underlined and the summation convention applies to the Index j. When a solution of these equations Is required In some bounded domain Q, with boundary f2, use Is generally made of an appropriate difference approximation. A new class of such approximations was Introduced and utilized In 1 and 2; It Is the purpose of this paper to establish the convergence of the solutions of such approximations to the solutions of equations (1) and (2) In Q. To our knowledge, the first convergence proof for a difference approximation to the complete system (l) and (2) was given by Krzhlvltskl and Ladyzhenskaya (see e.g. 3)- Their proof gives both more and less than the numerical analyst requires.
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