Page 91 -

2.3 The Stokes Equations 75

Note that for the interior Neumann problem, the modified integral equa-
tions will provide specific uniquely determined solutions of the integral equa-
tions, whereas the solution of the original Stokes Neumann problem still has
the nullspace a + b × x for n =3 and {a + b(x 2 , −x 1 ) } for n =2.

Finally, the second versions of the modified integral equations (II) in
Table 2.3.4 are often referred to as stabilized versions in scientific computing.
Clearly, the two versions are always equivalent Fischer et al [79].

2.3.3 The Incompressible Material — Revisited
With the analysis of the Stokes problems available, we now return to the
interior elasticity problems in Section 2.2.4 for almost incompressible mate-
rials, i.e., for small c ≥ 0, but restrict ourselves to the case that Γ is one
connected compact manifold (see also [143], and Steinbach [289]). The case
of Γ = L " Γ as in Theorem 2.3.2 is considered in [143].
=1
For the interior displacement problem, the unknown boundary traction σ
satisfies the boundary integral equation (2.2.40) of the first kind,
1
V e σ =( I + K e )ϕ on Γ. (2.3.37)
2
where the index e indicates that these are the operators in elasticity where
the kernel E e (x, y) can be expressed via (2.2.68). Then with the simple layer
potential operator V st of the Stokes equation and its kernel given in (2.3.10)
we have the relation
1 2c 1
V e = V st + V ∆ I (2.3.38)
1+ c 1+ c µ
where V ∆ denotes the simple layer potential operator (1.2.1) of the Laplacian.
Inserting (2.3.38) into (2.3.37) yields the equation
2
1
V st σ =(1 + c)( I + K e )ϕ − c V ∆ σ , (2.3.39)
2
µ
which corresponds to the equation (1) of the interior Stokes problem in
Table 2.3.3.
As was shown in Theorem 2.3.2, the solution of (2.3.39) can be decom-
posedinthe form

σ = σ 0 + αn with σ 0 · nds =0 and α ∈ IR . (2.3.40)
Γ
Hence,

2
1
V st σ 0 =(1 + c)( I + K e )ϕ − c V ∆ (σ 0 + αn) ,
2
µ
(2.3.41)

σ 0 · nds =0 .
Γ