Finite Element Methods for Parabolic Problems - Computational Methods for Quantitative Finance - page 36

Fig. 3.1 Reference element
shape functions on the
reference element K

Furthermore, for i, j = 1 , 2 there holds

a l (N j

K l ,N i K l )

( A l ) ij =

α(x)∂ x N j

K l N i K l d x

β(x)∂ x N j γ(x)N j

K l ∂ x N i K l + K l N i K l +

=

K l

h l
2

α K l (ξ ) 4

h l N j N i + β K l (ξ ) 2

h l N j N i +

γ K l (ξ )N j N i = d ξ,

K

where

α F K l (ξ ) , β F K j (ξ ) , γ F K l (ξ ) .

β K l (ξ )

α K l (ξ ) := := γ K l (ξ ) :=

For later purpose, it is useful to split the integral into three parts

( A l ) ij = ( S l ) ij + ( B l ) ij + ( M l ) ij , (3.22)

where

2
h l

K α K l N j N i d ξ, K β K l N j N i d ξ,

( S l ) ij = ( B l ) ij =

h l
2

γ K l N j N i d ξ. ( M l ) ij =

K

For general coefficients α(x) , β(x) and γ(x) , these integrals cannot be computed
exactly. Hence, they have to be approximated by a numerical quadrature

1

p

w (p)
j

f(ξ (p)
j

f(ξ) d ξ ≈ ),

− 1

j = 1

using a p -point quadrature rule with quadrature weights w (p)
j

and quadrature points

ξ (p)
j

1 ,...,p .
It remains to construct the stiffness matrix A . The idea is to express the global ba-
sis functions b i (x) in terms of the local shape functions. We have for i

∈ ( − 1 , 1 ) , j =

= 1 ,...,N ,

N K i + N K i + 1 , 0 (x) N K 1 , N + 1 (x) N K N + 1 .

= = =

b i (x)