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108 3. Finite Element Methods for Linear Elliptic Problems

However, we stress that the conditions of Theorem 3.15 are only suffi-
cient, since concerning the V -ellipticity, it might also be possible to balance
an indefinite addend by some “particular definite” addend. But this would
require conditions in which the constants C P and C F are involved.
Note that the pure Neumann problem for the Poisson equation

−∆u = f in Ω ,
(3.33)
∂ ν u = g on ∂Ω
is excluded by the conditions of Theorem 3.15. This is consistent with the
fact that not always a solution of (3.33) exists, and if a solution exists, it
obviously is not unique (see Exercise 3.8).
Before we investigate inhomogeneous Dirichlet boundary conditions, the
application of the theorem will be illustrated by an example of a natural
situation described in Chapter 0.
For the linear stationary case of the differential equation (0.33) in the
form

∇· (cu − K∇u)+ ˜ru = f
we obtain, by differentiating and rearranging the convective term,
−∇ · (K∇u)+ c ·∇u +(∇· c +˜r) u = f,

which gives the form (3.12) with r := ∇·c+˜r . The boundary ∂Ω consists
only of two parts Γ 1 and Γ 2 .Therein,Γ 1 an outflow boundary and Γ 2 an
inflow boundary; that is, the conditions
c · ν ≥ 0on Γ 1 and c · ν ≤ 0 on Γ 2
hold. Frequently prescribed boundary conditions are

−(cu − K∇u) · ν = −ν · cu on Γ 1 ,
−(cu − K∇u) · ν = g 2 on Γ 2 .

They are based on the following assumptions: On the inflow boundary Γ 2
the normal component of the total (mass) flux is prescribed but on the
outflow boundary Γ 1 ,on which in the extreme case K = 0 the boundary
conditions would drop out, only the following is required:

• the normal component of the total (mass) flux is continuous over Γ 1 ,
• the ambient mass flux that is outside Ω consists only of a convective
part,
• the extensive variable (for example, the concentration) is continuous
over Γ 1 , that is, the ambient concentration in x is also equal to u(x).

Therefore, after an obvious reformulation we get, in accordance with the
definitions of Γ 1 and Γ 2 due to (3.18), (3.19), the Neumann boundary