In this section, we introduce our BDDC preconditioner for problem (2.7) and describe the BDDC algorithm.
We first define a discrete harmonic operator associated with the rotated element: for any , let such that
here . Let . We define ℋ as a corresponding piecewise harmonic operator on the auxiliary space by .
In order to introduce our domain decomposition method, we decompose the auxiliary discrete space as follows:
where the space is a piecewise harmonic function space defined as
We define a space . The space is between and , and our BDDC preconditioner is mainly constructed on this space.
As we know, the technical aspect in DDMs is that the preconditioner includes a coarse problem which can enhance the convergence. In view of the characteristic of the space , we select the standard coarse space which is the rotated finite element space associated with the coarse partition , and it satisfies primal constraints on subdomain interfaces.
The substructure space with constraints is defined by
Denote . The coarse space and the product space play an important role in the description and analysis of our iterative method.
To present our BDDC preconditioner, we introduce several space transfer operators. Define an interpolation operator by
The intergrid transfer operator is defined by
Define an extension operator as
for any , ;
for any , ;
for any , satisfies (2.5).
Its transpose is defined by
Denote by , the corresponding transpose is defined by
We also need to define another prolongation operator as follows:
if , then ;
if , , then ;
if , , , then ;
if , it follows from (2.5) that can be obtained by the edge average values on associated mortar sides;
In what follows, we describe our BDDC preconditioning algorithm, we apply the basic framework of additive Schwarz method (or parallel subspace correction method ). From the decomposition (3.1), we only need to choose appropriate subspace solvers.
First of all, the coarse subspace solver is defined by
On each subdomain, similar operators and are defined, respectively, by
Remark 3.1 The bilinear form on the coarse space can be different from that on substructure space, here we only use the exact solvers. On each subdomain, we avoid the possible singularity of local subproblem and we need not modify the bilinear forms.
Now we define our BDDC preconditioner as
where , is the corresponding transpose defined by
Let be an operator from to defined by
and be the operators from to and defined, respectively, by
Then the BDDC preconditioned operator can be written as
We have the following main result.
The BDDC preconditioned operator