Non-overlapping Subdomain Methods

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## Non-overlapping Subdomain Methods

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:

1. inherits the symmetric positive definiteness of

.

2. is dense in general and computing it explicitly requires as many solves on each subdomain as there are points on each of its edges.

3. The condition number of

is

, an improvement over the

growth for

.

4. Given a vector

defined on the boundary vertices

of

, the matrix-vector product

can be computed according to

where

involves

independent subdomain solves using

.

5. The right hand side

can also be computed using

independent subdomain solves.

These properties make it possible to apply a preconditioned iterative method to (), which is the basic method in the nonoverlapping substructuring approach. We will also need some good preconditioners to further improve the convergence of the Schur system.

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

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 [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

.

Next: Further Remarks Up: Domain Decomposition Methods Previous: Overlapping Subdomain Methods

Jack Dongarra
Mon Nov 20 08:52:54 EST 1995