My current research interest is domain decomposition methods, in particular the establishment of a new theoretical framework for a new algorithm called Algorithm 3 (or OSM-D).
The numerical solution of partial differential equations from many industrial or academic problems often lead to quite large, sparse linear systems. Domain decomposition methods are general flexible iterative methods for solving such problems. These algorithms are divided into two classes, those that use overlapping domains, which are often referred to as Schwarz methods, and those that use nonoverlapping domains, which are called iterative substructuring methods. The classical overlapping multiplicative Schwarz algorithm which uses Dirichlet boundary conditions at each artificial interface is known to be successful for solving Poisson's problem. However, this algorithm is unsuccessful in solving Helmholtz's problem. To overcome the problems in the classical algorithm, a new family of domain decomposition methods, which uses discontinuous iterates for the solution of Helmholtz's equation, has been presented by Cai, Casarin, Elliott, and Widlund [1]. The new algorithm is quite effective for solving Helmholtz's equation in different contexts, but its analysis and implementation is complicated by the fact that it allows jumps in the iterates across the artificial interfaces.
The study of new algorithm has posed many interesting questions
related to how far and in what sense the classical theory of
domain decomposition methods can be extended
to this new setting, such as to what
extent there may be a counterpart of the geometric convergence factor
of classical Schwarz methods and of the energy estimate of the Robin
iteration methods in Lions [4].
It is also interesting to see what new phenomena occur and what new insight
this might lend to the classical algorithms.
In order to construct a general framework and fundamental theory of
Algorithm 3 (OSM-D),
my research has concentrated on Poisson's problem.
In particular, I am interested in how the classical theory
can be generalized.
1. Background of Algorithm 3 (OSM-D)
In [1], three algorithms for the solution of Helmholtz's
equation are presented.
They are based on the overlapping Schwarz methods and called Algorithm 1,
Algorithm 2, and Algorithm 3 in increasing order of sophistication.
Algorithm 1 is a classical Schwarz algorithm.
Algorithm 2 (OSM-C) improves on Algorithm 1
by using approximate Sommerfeld boundary conditions at each artificial
interface while maintaining continuity of the iterates.
As a new type of overlapping Schwarz methods, Algorithm 3 is
constructed from Algorithm 2 and allows for discontinuities at each artificial
interface.
The new algorithm is inspired by the
thesis of Després' [2] and it can be considered as
an overlapping version of Després methods.
The main topic of my research, Algorithm 3 (OSM-D) for
Poisson's equation, can also be considered
as an overlapping version of the Robin iteration methods
found in Lions [4].
2. Formulation of Algorithm 3 (OSM-D)
The first questions are related to how to extend the relevant
notations of the classical algorithm to a general setting
and how to formulate Algorithm 3 (OSM-D) carefully to understand
the differences between the new and classical algorithms. One
focus of my research is the extension
and interpretation of a new algorithm inside the classical theory.
The notations in [1]
are revised and extended to a new setting for Poisson's equation.
The variational formulation of Algorithm 3 (OSM-D) is
derived from Algorithm 2 (OSM-C). The discontinuity of the iterates of
the new algorithm, which is the fundamental
distinction from the classical algorithms, is implemented by allowing
multiple values
on the artificial interfaces.
To analyze this important property, I use the
Saddle-point approach which is also used for formulating the
Finite Element Tearing and Interconnecting (FETI)
method by Farhat and
Roux [3]. I show that
Algorithm 3 (OSM-D) can be interpreted as a
Block Gauss-Seidel method with dual and primal values;
a new dual value can be computed from given primal values and the dual values
will then be used to compute a new primal value.
I also show that the algorithm, induced from the variational formulation,
is identical to that derived from the saddle-point approach.
This means that the dual values, which are not computed explicitly,
can be viewed as an
essential
part of the new algorithms.
3. Algebraic Properties of Algorithm 3 (OSM-D)
A new overlapping Schwarz algorithm naturally poses
questions about its algebraic structure.
In classical theory,
each fractional step is symmetric with respect to the 
inner product.
One of the important results of my research is nonsymmetry of the fractional
steps of Algorithm 3 (OSM-D).
Therefore, it is impossible
to apply the classical Conjugate Gradient methods.
The algebraic systems of primal values can be reduced to those of
dual values. In the two overlapping subdomain case,
the dual value systems results in a block 
-cycle matrix.
I analyze the structure of the subblock matrices algebraically and
check their numerical behavior.
4. Convergence Theory of Algorithm 3 (OSM-D)
Convergence theory is the most interesting question of my research. I start my study of the convergence theory with two overlapping strips with Dirichlet boundary condition. I show that the rates of convergence on the two nonoverlapping parts is better than that of the Robin iteration methods. Inspired by [5], I show that the convergence is geometric for the case of several overlapping strips. I extend the results to a general quadrilateral by using conformal mapping.
The concept of cross point related to the overlapped part of at least three subdomains, is a crucial part of the new algorithm. Without cross points, Algorithm 2 (OSM-C) and Algorithm 3 (OSM-D) have identical updated values. I have studied the convergence theory with general (Robin) boundary condition, general geometry, and cross points. Numerical experiments show that Algorithm 2 (OSM-C) diverges in some cases with cross points but that Algorithm 3 (OSM-D) converges with any combination of conditions. From the point of view of a Block Gauss-Seidel method, Algorithm 2 (OSM-C), which does not retain old values on the artificial interfaces, brings about a loss of information for the dual values.
From an observation of the structure of the submatrices of all atomic subregions, I have also discovered that the atomic subdomains corresponding to cross points always have positive definite local matrices regardless of shape and the number of overlapped subdomains. In a special geometry, I have proven geometric convergence of the dual values, which implies convergence of primal values, in the two overlapping subdomain case with any Robin boundary condition. I have also shown that this idea can be extended to the general two overlapping subdomain cases. In a next step, I have extended my idea to the general case with cross points. I have proved that the error vectors on regions with cross points can decrease geometrically under plausible conditions of the dual values of the neighboring atomic subdomains, which can be induced from the algebraic structure of the local matrices corresponding to the subdomains. I have also considered the convergence of the entire error vector using the previous results.
Another interesting question in my research is multi level variants of
Algorithm 3 (OSM-D).
The numerical results show that two-level methods with the Robin boundary
condition, which is close to the Neumann boundary condition on the artificial
interfaces
can even diverge
in some
cases.
5. Conclusion
In the future, I plan to explore some unsolved questions from my current research. I also hope to extend my research and answer some more ambitious questions.