Finite Element Methods for Parabolic Problems - Computational Methods for Quantitative Finance - page 41

H 0 (G) , and if V N = S 1

T

Ch − 2 as h

V =

, we obtain λ A ∼

↓

0. Hence, we get stabil-

ity provided the CFL type stability condition

θ< 1
2

Ch 2 /( 1 k ≤ − 2 θ), 0 ≤

1

holds. For

2 ≤

θ ≤ 1, the θ -scheme is stable without any time step restriction.

3.6 Error Estimates

Let u m (x) u(t m ,x) , u N = be as in ( 3.14 ) and assume V N consists of linear finite

S 1
T

elements, i.e. V N = .For m = 0 ,...,M − 1, we want to estimate the error

e N (x) u m (x) u N (x) . Therefore, we split the error

e N = (u m

:= −

− I N u m ) + ( I N u m − u N ) =: η m + ξ N , (3.34)

where I N : V → V N is the interpolant as defined in ( 3.24 ). For a fixed time point

t m , η m (x) = u(t m ,x) − ( I N u)(t m ,x) ∈ V is a consistency error for which we now

give an error estimate.

3.6.1 Finite Element Interpolation

We prove error estimates of the interpolation error u − I N u .

Proposition 3.6.1 Let I N : V → V N be the interpolant as defined in ( 3.24 ). Then ,

the following error estimates hold :

N + 1

h 2 ( − n)
i

− I N u) (n)

2

u ()

2

(u

L 2 (G) ≤

C

L 2 (K i ) ,n

= 0 , 1 , = 1 , 2 . (3.35)

i = 1

In particular , if the mesh is uniform , i . e . h i = h ,

Ch − n − I N u) (n) u ()

(u L 2 (G) ≤ L 2 (G) ,n = 0 , 1 , = 1 , 2 . (3.36)

u − c ∈ H 1 (G) for c = 1

Proof Consider G = ( 0 , 1 ) and u ∈ H 2 (G) . Then,

0 u .
By the Poincaré inequality ( 3.4 ), which also holds for the space H 1 (G) (see ( 3.5 )),

u − c L 2 (G) ≤ C u L 2 (G) . (3.37)

With

x

I N

u := u( 0 ) + c d x = u( 0 ) + cx,

0