Finite Difference Domain Decomposition for a Singularly Perturbed
Parabolic Problem

Igor Boglaev
Mathematics, Institute of Fundamental Sciences, Massey University
Private Bag 11-222, Palmerston North, New Zealand
e-mail: I.Boglaev@massey.ac.nz

Iterative domain decomposition algorithms for the solution of
singularly perturbed elliptic and parabolic problems have been
proposed in [1], [2]. These algorithms for parabolic problems are
based on a combination of an implicit time discretization and domain
decomposition methods. On each time step, this approach reduces a
given parabolic problem to a sequence of elliptic problems on
appropriate  subdomains, where regions of rapid change of the
solution (boundary layers) are localised in subdomains.
The algorithms from [1], [2] have been constructed and analysed in
continuous forms, i.e. without resort to a spatial discretization in
subdomains. In [3] finite domain decomposition methods for one
dimensional singularly perturbed parabolic problems have been
investigated. Uniformly convergent methods on a special graded mesh
of Bakhvalov's type [4] have been examined.
In this paper, we present finite difference domain decomposition
methods based on a mesh of Shishkin' s type [5] for solving two
dimensional singularly perturbed parabolic problem. These meshes
allow us to decompose the computational domain into subdomains
outside the boundary layers and inside them as well. Our purpose is
to construct domain decomposition algorithm s based on decomposition
of boundary layers, which are suitable for parallel computing.
Firstly, we consider the undecomposed algorithm from [5] which
exhibits an uniform in a small parameter convergence.
We present a fully implicit in time domain decomposition algorithm,
where implicit time difference schemes are used in main and
interfacial subdomains. We consider two domain decomposition, the
first one is decomposition with interfacial subdomains located
outside the boundary layers and the second one with interfacial
subdomains located inside the boundary layers.
Finally, we present an implicit-explicit algorithm, where explicit
difference schemes are applied outside the boundary layers.

References
1.  I.P. Boglaev, Iterative algorithms of domain decomposition for
the solution of a quasilinear elliptic problem, J. of Computational
and Applied Mathematics 80, 299-316, (1997).
2.  I.P. Boglaev, V.V. Sirotkin, Iterative domain decomposition
algorithms for the solution of singularly perturbed parabolic
problems,  Computers Math. Applic. 31, 10, 83-100, (1996).
3.  I.P. Boglaev, Finite difference domain decomposition algorithms
for a parabolic problem with boundary layers, Preprint Series B:
97/02,  FIMS, Massey University, New Zealand,  1997.
4.  N.S. Bakhvalov, On the optimization of methods for solving
boundary value problems in the presence of a boundary layer,  Zh.
Vychisl. Mat. i Mat. Fiz. 9, 841-859 (1969).
5.  J.J.H. Miller, E. O'Riordan, G.I. Shishkin, Fitted numerical
methods for singular perturbation problems, World Scientific,
Singapore, 1996.