Finite Element Methods for Parabolic Problems - Computational Methods for Quantitative Finance - page 38

Fig. 3.2 Function u(x) and
its piecewise linear
approximation ( I N u)(x)

where the load vector l is given by

M φ − l (t) = f (t) − A φ ,

2 denoting the coefficient vector of φ .For
Neumann boundary conditions, i.e. ∂ x u(t, a)

(φ 1 (t), 0 ,..., 0 ,φ 2 (t)) ∈ R

N +

with φ =

= φ 1 (t) , ∂ x u(t, b) = φ 2 (t) , the bound-

ary degrees of freedom must be kept.

3.4.2 Initial Data

H 1 (G)
(e.g. u 0 being the payoff of a put or call option), we can use nodal interpolation ,i.e.
u N, 0 = I N u 0 .For u ∈ H 1 (G) , define the nodal interpolant

In the discretization ( 3.14 ), we need an approximation of u 0 in V N .If u 0 ∈

I N u ∈ S 1

by

T

N + 1

( I N u)(x) := u(x i )b i (x), (3.24)

i = 0

as illustrated in Fig. 3.2 . Since b i (x j ) = δ ij , we get ( I N u)(x i ) = u(x i ) .
For u

H 1 (G) , the nodal value u(x i ) is well defined due to Theorem 3.1.4 .

∈

H 1 (G) S 1

T

Lemma 3.4.2 The interpolation operator defined in ( 3.24 ) is

bounded , i . e . there exists a constant C ( which is independent of N ) such that

I N : →

H 1 (G). I N u H 1 (G) ≤ C u H 1 (G) , ∀ u ∈ (3.25)

L 2 (G) (e.g. u 0 being the payoff of a digital option), we
cannot use interpolation, but need the L 2 -projection of u 0 on V N ,i.e. u N, 0 = P N u 0 .
For u

If we only have u 0 ∈

L 2 (G) ,the L 2 -projection of u on V N is defined as the solution of

∈

G P N uv N d x

= uv N d x, ∀ v N ∈ V N . (3.26)

G

Using the basis b i , i = 0 ,...,N + 1, of V N we can obtain the coefficient vector u N

of P N u 0 as the solution of the linear system

M u N = f , with f i = ub i d x, i = 0 ,...,N + 1 .

G