Wavelet Methods - Computational Methods for Quantitative Finance - page 170

S r (

∈ M

Replacing u L by a function U ,V L ) and integrating by parts in each J m ,we

obtain with w m = w(t m )

M

(U, w)

(U ,w) d t

t t m − 1 −

(u L ,w ) d t − =− |

J J m

m = 1

M − 1

(U ,w) d t ]] m ,w m ) (U 0

+

,w 0 ).

= + ( [[ U +

J

m = 1

Therefore, we obtain the fully discrete scheme: Find U ∈ S r ( M ,V L ) such that for
all W S r (

∈ M ,V L )

(U ,W)

a(U,W) d t

M − 1

]] m ,W m (U 0

+

,W 0
+

(u L, 0 ,W 0 + + ( [[ U ) + ) = ).

+ +

J

m = 1

(12.10)
The solution operator of the parabolic problem generates a holomorphic semigroup.
Therefore, the solution u(t) is analytic with respect to t for all t> 0. However,
due to the non-smoothness of the initial data, the solution may be singular at t = 0.

By the use of the so-called geometric time discretization, the low regularity of the
solution at t = 0 can be resolved.

M
m

Definition 12.3.1 We call a partition M M,γ ={ J m } of the time interval J =

= 1

( 0 ,T) ,0 <T < ∞ , geometric with M time steps J m = (t m − 1 ,t m ) , m = 1 ,...,M ,

and grading factor γ ∈ ( 0 , 1 ) if

Tγ M − m , t 0 = 0 , t m = 1 ≤ m ≤ M.

A polynomial degree vector r = (r 1 ,...,r M ) is called linear with slope μ> 0on

M M,γ if r 1 = 0 and r m = μm , m = 2 ,...,M , where μm = max { q ∈ N 0 : q ≤

}

μm .

We have the following a priori error estimate on the hp -dG scheme [123, Theo-
rem 3].

∈ H s (G) ,0 <s Theorem 12.3.2 Let u 0 1 and the assumptions of Theo-

rem 12.2.4 be fulfilled . Then , there exist μ 0 ,m 0 > 0 such that for all geometric par-
titions

≤

M M,γ with M ≥ m 0 | log h | , and all polynomial degree vectors r on M M,γ

with slope μ>μ 0 , the fully discrete solution U obtained by ( 12.10 ) satisfies

u(T ) − U(T) L 2 (G) ≤ Ch p , (12.11)

where C> 0 is a constant independent of mesh width h , and u is the solution of the
parabolic problem ( 12.5 ).