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The easiest way to describe this approach is through
matrix notation.
The set of vertices of can be divided into two groups.
The set of interior vertices
of
all
and the set of vertices
which lies on the boundaries
of the coarse triangles
in
.
We shall re-order
and
as
and
corresponding to this partition.
In this ordering, equation (
) can be written
as follows:
We note that since the subdomains are uncoupled by the boundary vertices,
is block-diagonal
with each block
being the stiffness matrix corresponding
to the unknowns belonging to the interior
vertices of subdomain
.
By a block LU-factorization of , the system in (
)
can be written as:
where
is the Schur complement of
in
.
By eliminating in (
), we arrive at
the following equation for
:
We note the following properties of this Schur Complement system:
We shall first describe
the Bramble-Pasciak-Schatz preconditioner [35].
For this, we need to further decompose into two non-overlapping
index sets:
where
denote the set of
nodes
corresponding to the vertices
's of
, and
denote the set of
nodes
on the edges
's
of the coarse triangles in
, excluding
the vertices
belonging to
.
In addition to the coarse grid interpolation and restriction
operators defined before,
we shall also need a new set of interpolation and restriction
operators for the edges
's.
Let
be the pointwise restriction operator
(an
matrix, where
is the number
of vertices on the edge
)
onto the edge
defined by its action
if
but is zero otherwise.
The action of its transpose extends by zero a function
defined on
to one defined on
.
Corresponding to this partition of ,
can be written in
the block form:
The block can again be block partitioned, with most of
the subblocks being zero. The diagonal blocks
of
are the
principal submatrices of
corresponding to
.
Each
represents the coupling of nodes on interface
separating two neighbouring subdomains.
In defining the preconditioner, the action of is
needed. However, as noted before, in general
is a dense
matrix which is also expensive to compute, and even if we had it, it
would be expensive to compute its action (we would need to compute its
inverse or a Cholesky factorization). Fortunately, many efficiently
invertible approximations to
have been proposed in the
literature (see Keyes and Gropp [136]) and we shall use these
so-called interface preconditioners for
instead.
We mention one specific preconditioner:
where
is an
one dimensional
Laplacian matrix, namely a tridiagonal matrix with 2's down
the main diagonal and -1's down the two off-diagonals,
and
is taken to be some average of the coefficient
of (
) on the edge
.
We note that since the eigen-decomposition of
is known
and computable by the Fast Sine Transform, the action of
can be efficiently computed.
With these notations, the Bramble-Pasciak-Schatz preconditioner
is defined by its action on a vector defined on
as follows:
Analogous to the additive Schwarz preconditioner, the computation of each term consists of the three steps of restriction-inversion-interpolation and is independent of the others and thus can be carried out in parallel.
Bramble, Pasciak and Schatz [35] prove that the condition
number of is bounded by
. In
practice, there is a slight growth in the number of iterations as
becomes small ( i.e., as the fine grid is refined) or as
becomes large ( i.e., as the coarse grid becomes coarser).
The growth is due to the coupling of the unknowns on the
edges incident on a common vertex
, which is not accounted for in
. Smith [187] proposed a vertex space
modification to
which explicitly accounts for this coupling
and therefore eliminates the dependence on
and
. The idea is
to introduce further subsets of
called vertex spaces
with
consisting of a small set of vertices on
the edges incident on the vertex
and adjacent to it. Note that
overlaps with
and
. Let
be the principal
submatrix of
corresponding to
, and
be
the corresponding restriction (pointwise) and extension (by zero)
matrices. As before,
is dense and expensive to compute and
factor/solve but efficiently invertable approximations (some using
variants of the
operator defined before) have been developed
in the literature (see Chan, Mathew and
Shao [50]). We shall let
be such a
preconditioner for
. Then Smith's Vertex Space
preconditioner is defined by:
Smith [187] proved that the condition number
of is bounded independent of
and
, provided
there is sufficient overlap of
with
.