Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 233

Fig. 51. Decomposition of a nodal basis function on the coarse grid (top) in
terms of nodal basis functions on the fine grid (bottom)

− a(u ,k, 1 ,w) . In particular, d =

0if
u ,k, 1 is a solution at level . As in the derivation of (II.4.4), we successively insert
w = ψ − 1

Here d is defined by (d, w) 0 : = (f, w) 0

for j = 1 , 2 ,...,N − 1 , and immediately take into account (1.10):

j

r ji a(u ,k, 1 + v, ψ i ) = r ji (f, ψ i ) 0 ,j = 1 , 2 ,...N − 1 .

i i

= t x ,k, 1 ψ t .Nextweset v = s y − 1 We recall that u ,k, 1 ψ − 1 , and return to

t

s s

the basis of S :

r ji (f, ψ i ) 0 −

,

a(ψ t ,ψ i )x ,k, 1 r ji a(ψ s ,ψ i )r st y − 1 =

t t

t t

i,s i

j = 1 , 2 ,...,N − 1 . ( 1 . 11 )

The expression in the square brackets is just the i -th component of the residue
d defined in (1.9). Thus, (1.11) is the componentwise version of the equation
rA r t y − 1

= rd , and (1.9) follows with p = r t .

For completeness we note that the vector representation of the approximate
solution after the coarse-grid correction (1.8) is

x ,k, 2 = x ,k, 1 + py − 1 .

( 1 . 12 )

In practice we usually compute the prolongation and restriction matrices via
interpolation. Let { ψ i } be a nodal basis for S . Then we have N points z t with

ψ i (z t ) = δ it , t = 1 , 2 ,...,N .

For every v ∈ S , v = i v(z i )ψ i , and so ψ − 1 = i ψ − 1 (z i )ψ i for the basis

j j

functions of S − 1 . Comparing coefficients with (1.10), we get

r ji = ψ − 1 (z i ).

( 1 . 13 )

j

This is a convenient description of the restriction matrix.