S u + k B p = g, B^T u = 0

Rolf Reinelt

In the book "Efficient Solvers for Incompressible Flow Problems" by Turek the resulting algebraic equation system of the discretised Navier-Stokes equations is given as

S u + k B p = g

B^T u = 0

where

u = (u1,u2,...,um,v1,v2,...,vm)^T

and

p = (p1,p2,...,pn)^T

are vectors and S and B are matrices. An equivalent form for steady flows would be

S u + B1 p = g1

S v + B2 p = g2

B1^T u + B2^T v = 0

where

u = (u1,u2,...,um)^T

v = (v1,v2,...,vm)^T

The crucial point is that we can use B^T in the continuity equation. My questions are:

1) Are there other books that use this formulation?

2) Is it true that this formulation is valid for finite difference, finite volume and finite element methods?

3) Are the known and unknown boundary values supposed to be part of the vectors u,v,p?

4) When using finite volume method, what are the restrictions that the formulation remains valid?

5) If m=n, i.e. the number of pressure nodes = number of u-velocity nodes, can the continuity equation be written as B1 u + B2 v = 0 ?

Kind regards, Rolf