Finite Element Methods for Parabolic Problems - Computational Methods for Quantitative Finance - page 44

The stability of the θ -scheme, Proposition 3.5.1 ,gives

Corollary 3.6.8 Under the assumptions of Proposition 3.5.1 ,

M − 1 M − 1

ξ N

2

ξ m + θ
N

2

ξ N

2

r m

2
∗

L 2 (G) +

C 1 k

a ≤

L 2 (G) +

C 2 k

0

. (3.46)

m = 0 m =

r m To prove Theorem 3.6.5 , it is therefore sufficient to estimate the residual ∗ .

Proof of Theorem 3.6.5
(i) Estimate of r 1 : for any v N ∈ V N ,wehave

(r 1 ,v N ) k − 1 (u m + 1 u m ) u m + θ

| |≤ − −˙ ∗ v N a .

With

k − 1 t m + 1
t m

k − 1 (u m + 1 u m ) u m + θ − −˙ = (s − ( 1 − θ)t m + 1 − θt m ) u d s, ¨

we get

k − 1 t m + 1
t m

k − 1 (u m + 1 u m ) u m + θ − −˙ ∗ ≤ | s − ( 1 − θ)t m + 1 − θt m |¨ u ∗ d s

C θ k 2 t m + 1
t m

d s

1
2

2
∗

≤ ¨ u(s) .

(ii) Estimate of r 2 : for any v N ∈ V N ,

C k − 1 (u m + 1 u m ) ∗ (r 2 ,v N ) u m ) − I N (u m + 1 | |≤ − − v N a

Ck − 1 u(s) d s ∗ − I N ) t m + 1

t m

= (I ˙ v N a

Ck − 1 t m + 1
t m

≤ ˙ u − I N ˙ u ∗ d s v N a .

(iii) Estimate of r 3 : using the continuity of a( · , · ) ,

(r 3 ,v N ) u m + θ − I N u m + θ | |≤ C a v N a .

We have proved that for every m = 0 , 1 ,...,M − 1,

Ck t m + 1
t m

r m

2 2

∗

∗ ≤

¨ u(x) d s

Ck − 1 t m + 1
t m

2
∗

u m + θ − I N u m + θ

2

+ ˙ u − I N ˙ u d s + C

a .