Finite Element Methods for Parabolic Problems - Computational Methods for Quantitative Finance - page 28

L 2 (G) , then u does not necessarily exist in the classical sense, but we may
define u to be the linear functional

∈

If u

u ∗ (ϕ) uϕ d x, C 0 (G). =− ∀ ϕ ∈

G

This functional is said to be a generalized or weak derivative of u . When u ∗
is bounded in L 2 (G) , it follows from Riesz representation theorem (see Theo-
rem 2.1.1) that there exists a unique function w

L 2 (G) such that u ∗ (ϕ)

∈ = (w, ϕ)

for all ϕ ∈ L 2 (G) , in particular

uϕ d x C 0 (G). − = wϕ d x, ∀ ϕ ∈

G G

We then say t ha t the weak derivative belongs to L 2 (G) and write u = w . In particu-
lar, if u C 1 (G) , the generalized derivative u coincides with the classical derivative

u . In a similar way, we can define weak derivatives D n u of higher order n

∈

∈ N .

Definition 3.1.1 The linear functional D n u , n

∈ N is a weak derivative of u if

1 ) n

D n uϕ d x uD n ϕ d x, C 0 (G). = ( − ∀ ϕ ∈

G G

We can now define the spaces H m (G) .

. H m (G) is the space of all functions whose weak partial Definition 3.1.2 Let m ∈ N

m belong to L 2 (G) ,i.e.

derivatives of order ≤

H m (G) L 2 (G) D n u L 2 (G) for n ={ u ∈ : ∈ ≤ m } .

We equip H m (G) with the inner product

m

(D n u, D n v) L 2 (G) , (u, v) H m (G) =

n = 0

and the corresponding norm

m

2

D n u

2

u

H m (G) =

(u, u) H m (G) =

0

L 2 (G) .

n =

We sometimes omit the (G) if the domain is clear from the context. H m (G) is
complete and thus a Hilbert space. The space H m (G) is an example of a more
general class of function spaces, called Sobolev spaces .

. W m,p (G) is the space of all functions whose

Definition 3.1.3 Let p ∈ N ∪{∞}

m belong to L p (G) ,i.e. weak partial derivatives of order ≤

W m,p (G) ={ u ∈ L p (G) D n u ∈ L p (G) for n ≤ m } . :