Wavelet Methods - Computational Methods for Quantitative Finance - page 168

=

corresponding to level . Then, ( 12.3 )for s α/ 2 implies that

2

u , A u ∼ u

H α/ 2 (G) ∼

u, D u .

D 1 / 2 u ,wehave Written in terms of u =

u, D − 1 / 2 AD − 1 / 2

2 .

u ∼| u | (12.9)

And so we obtain that the condition number κ( D − 1 / 2 AD − 1 / 2 ) is bounded, indepen-
dent of the level L .

Example 12.2.5 We compute the condition number for several operators with order
ρ on various number of degrees of freedom N in Fig. 12.4 . It can be seen that in the
hat basis the condition numbers grow like

(N ρ ) . Using the multi-scale wavelet
basis with preconditioning, the condition numbers are bounded, independent of N .

O

We next address the time-discretization to obtain a fully discrete algorithm. As
before, we could use a θ -scheme to perform the timestepping. However, since the
parabolic problem generates an analytic semigroup, we present now a high-order,
discontinuous Galerkin ( hp -dG) timestepping scheme.

12.3 Discontinuous Galerkin Time Discretization

M
m

∞ ∈ N M ={

J m }

=

For 0 <T < and M ,let be a partition of J ( 0 ,T) into M

= 1

subintervals J m = (t m − 1 ,t m ) , m = 1 ,...,M , with

0 = t 0 <t 1 <t 2 < ··· <t M = T.

Moreover, denote by k m = t m − t m − 1 the length of J m .For u ∈ H 1 ( M ,V L ) ={ v ∈
L 2 (J, V L ) H 1 (J m ,V L ), m : v | J m ∈ = 1 ,...,M } , define the one-sided limits

u m

+ =

lim u(t m + s), m = 0 ,...,M − 1 ,

0 + s →

u m

− =

lim u(t m − s), m = 1 ,...,M,

0 +

s →

and the jumps

u m u m

−

[[ u ]] m =

+ −

, m = 1 ,...,M − 1 .

To each time step J m , we associate an approximation order r m ≥ 0. The orders are

collected in the degree vector r (r 1 ,...,r M ) . We introduce the following space of

functions which are discontinuous in time:

=

S r ( L 2 (J, V L ) S r m (J m ,V L ), m

M ={ ∈ : | J m ∈ = }

,V L ) u u 1 ,...,M ,

where S r m (J m ) denotes the space of polynomials of degree at most r m on J m .