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CHAPTER 3. MULTIGRID METHODS 46
cells we proceed recursively as follows. Assume for a set Λ(n) that the interpolation on
the unknowns in ∂Λ(n) has been fixed, 1 and we want to calculate the interpolation on the
nodes in Λ(n) \ ∂Λ(n). For that we build the local stiffness matrix of Ω(n) (consisting
of the element stiffness matrices of elements in Ω(n)) with the underlying partitioning
(Ω(n) \ ∂Λ(n)) ∪ ∂Λ(n)
( )
Kii K
K Ω(n) =
ib } Ω(n) \ ∂Λ(n)
K bi K bb } ∂Λ(n)
(i stands for interior, b for boundary) and perform local energy minimization:
( ) ( )
find u i such that (u T i u T Kii K
b )
ib ui
is minimized, with given u
K bb u b ,
b
K bi
with the result (for symmetric, positive definite K Ω(n) )
Now we can set
(P F C ) j,k =
where ˜P j is the localized version of P l
l+1 .
[
u i = −K −1
ii K ib u b .
( ( ) )
0 ←
−K −1
vertices of Λ(j) \ {k}
ii K ib
˜P j
1 ← k .
Remark 3.12. The drawback of this method is a possibly expensive set-up phase, as many
local minimization problems have to be solved and one has to save all the element stiffness
matrices. Approaches to overcome this difficulty can be found in [HV01].
Remark 3.13. A very nice property of element agglomerating AMGe is the complete information
about grid topology on the coarse levels. This could be utilized in various ways, so
e.g. stability analysis for saddle point problems can be performed nearly as in a geometric
context (c.f. Section 4.1.3) or as another example one could use the information to
construct some FAS-like schemes 2 for nonlinear problems, which is done in [JVW02].
∂Λ(j)
]
j
,
What we have not specified yet is how to construct the coarse agglomerates.
possibility for that is the following algorithm.
One
1 The ‘boundary’ ∂Λ(n) is defined straightforward: if Λ(n) is a face then ∂Λ(n) are those nodes of Λ(n)
which belong to more than one face, if Λ(n) is an agglomerated element then ∂Λ(n) is the union of faces
of this element.
2 FAS . . . full approximation storage, a multigrid method which is capable of solving nonlinear problems,
developed by Brandt [Bra77].