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:
inherits the symmetric positive definiteness of
.
is dense in general and computing it explicitly requires as many solves on each subdomain as there are points on each of its edges.
is
, an improvement over the
growth for
.
defined on the boundary vertices
of
, the matrix-vector product
can be computed according to
where
involves
independent subdomain solves using
.
can also be computed using
independent subdomain solves.
We shall first describe the Bramble-Pasciak-Schatz preconditioner [36]. 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 neighboring 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
's down the main diagonal and
's down the two off-diagonals, and
is taken to be some average of the coefficient
. 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 [36] 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 [191] 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 invertible approximations (some using variants of the
operator defined before) have been developed in the literature (see Chan, Mathew and Shao [52]). We shall let
be such a preconditioner for
. Then Smith's Vertex Space preconditioner is defined by:
Smith [191] proved that the condition number of
is bounded independent of
and
, provided there is sufficient overlap of
with