Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 264

Algebraic Description of Space Decomposition Algorithms

The finite element spaces S may be recursively constructed

S 0 = W 0 ,
S = S − 1 ⊕ W , ≥ 1 , ( 5 . 7 )

S = S L .

The finite element solution on the level is related to the operator A : S → S
defined by

(A u, w) = a(u, w) for all w ∈ S . ( 5 . 8 )

Moreover A : = A L . The corresponding Ritz projector P : S → S satisfies

a(P u, w) = a(u, w) for all w ∈ S .

( 5 . 9 )

We note that the discussion below holds for any inner product ( · , · ) in the Hilbert
space S , but we will refer to the L 2 inner product or the 2 inner product when
we deal with concrete examples. We recall that the L 2 -norm is equivalent to the
2 -norm of the associated vector representations, and the smoothing procedures
refer to L 2 -like operators. Therefore, we will use also the L 2 -orthogonal projectors
Q : S → S ,

(Q u, w) = (u, w) for all w ∈ S . ( 5 . 10 )

It follows that

A P = Q A. ( 5 . 11 )

Indeed, for all w ∈ S we obtain from (5.8)-(5.10) the equations (A P u, w) =
a(P u, w) = a(u, w) = a(u, Q w) = (Au, Q w) = (Q Au, w) . Since A P and
Q A are mappings into S , this proves (5.11).
Assume that u is an approximate solution of the variational problem in S , and

let A u ˜ − f be the residue. The solution of the variational problem in the subset

u + A − 1 Q (f − Au) . Therefore, the correction by the exact solution of
the subproblem for the level is

u + S is

A − 1 Q (f − Au).

Since its computation is too expensive in general, the actual correction will be
obtained from a computation with an approximate inverse B − 1 , i.e., the real cor-

rection will be

B − 1 Q (f − Au).

( 5 . 12 )

u + B − 1 Q (f − Au). For convenience, we will assume The correction turns u into

that

A ≤ B , ( 5 . 13 )