Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 28

§ 1. Sobolev Spaces

n

In the following, let be an open subset of R with piecewise smooth boundary.

The Sobolev spaces which will play an important role in this topic are built
on the function space L 2 () . L 2 () consists of all functions u which are square-
integrable over in the sense of Lebesgue. We identify two functions u and v
whenever u(x)

, except on a set of measure zero. L 2 () becomes
a Hilbert space with the scalar product

= v(x) for x ∈

(u, v) 0 : = (u, v) L 2 = u(x)v(x)dx ( 1 . 1 )

and the corresponding norm

(u, u) 0 .

u 0 = ( 1 . 2 )

∂ α u in L 2 () 1.1 Definition. u ∈ L 2 () possesses the (weak) derivative v =

provided that v ∈ L 2 () and

1 ) | α | (∂ α φ,u) 0 C 0 (). (φ, v) 0 = ( − for all φ ∈ ( 1 . 3 )

Here C ∞ () denotes the space of infinitely differentiable functions, and
C 0 () denotes the subspace of such functions which are nonzero only on a
compact subset of .

If a function is differentiable in the classical sense, then its weak derivative
also exists, and the two derivatives coincide. In this case (1.3) becomes Green's
formula for integration by parts.

The concept of the weak derivative carries over to other differential operators.
For example, let u ∈ L 2 () n be a vector field. Then v ∈ L 2 () is the divergence

of u in the weak sense, v = div u for short, provided (φ, v) 0 =− ( grad φ,u) 0 for

all φ ∈ C 0 () .