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100 3. Finite Element Methods for Linear Elliptic Problems

3.2 Elliptic Boundary Value Problems of Second
Order

In this section we integrate boundary value problems for the linear, sta-
tionary case of the differential equation (0.33) into the general theory of
Section 3.1.
Concerning the domain we will assume that Ω is a bounded Lipschitz
domain.
We consider the equation
(Lu)(x):= −∇ · (K(x)∇u(x)) + c(x) ·∇u(x)+ r(x)u(x)= f(x)for x ∈ Ω
(3.12)
with the data
d
K :Ω → R d,d , c :Ω → R , r, f :Ω → R.

Assumptions about the Coefficients and the Right-Hand Side
For an interpretation of (3.12) in the classical sense, we need
¯
∂ i k ij ,c i ,r ,f ∈ C(Ω) , i, j ∈{1,... ,d} , (3.13)
2
and for an interpretation in the sense of L (Ω) with weak derivatives, and
2
hence for a solution in H (Ω),
2
∂ i k ij ,c i ,r ∈ L (Ω) ,f ∈ L (Ω) , i, j ∈{1,... ,d} . (3.14)
∞
Once we have obtained the variational formulation, weaker assumptions
about the smoothness of the coefficients will be sufficient for the verifica-
tion of the properties (3.2)–(3.4), which are required by the Lax–Milgram,
namely,
2
k ij ,c i , ∇· c,r ∈ L (Ω) ,f ∈ L (Ω) , i, j ∈{1,... ,d} ,
∞
(3.15)
∞
and if |Γ 1 ∪ Γ 2 | d−1 > 0 , ν · c ∈ L (Γ 1 ∪ Γ 2 ) .
Here we refer to a definition of the boundary conditions as in (0.36)–(0.39)
(see also below). Furthermore, the uniform ellipticity of L is assumed: There
exists some constant k 0 > 0 such that for (almost) every x ∈ Ω,
d

2 d
k ij (x)ξ i ξ j ≥ k 0 |ξ| for all ξ ∈ R (3.16)
i,j=1
(that is, the coefficient matrix K is positive definite uniformly in x).
Moreover, K should be symmetric.
If K is a diagonal matrix, that is, k ij (x)= k i (x)δ ij (this is in particular
the case if k i (x)= k(x)with k :Ω → R,i ∈{1,...,d},where K∇u
becomes k∇u), this means that
(3.16) ⇔ k i (x) ≥ k 0 for (almost) every x ∈ Ω , i ∈{1,...,d} .