|dc.contributor.advisor||Loisel, Doctor Sebastien||
|dc.description.abstract||In modern science and engineering there exist many heterogeneous problems, in which
the material under consideration has non-uniform properties. For example when considering
seepage under a dam, water will flow at vastly different rates through sand and stone.
Mathematically this can be represented as an elliptic boundary value problem that has a
large jump in coefficients between subdomains. The optimised Schwarz method and the related
two-Lagrange multiplier method are non-overlapping domain decomposition methods
that can be used to numerically solve such boundary value problems.
These methods work by solving local Robin problems on each subdomain in parallel,
which then piece together to give an approximate solution to the global boundary value
problem. It is known that with a careful choice of Robin parameter the convergence of
these methods can be sped up.
In this thesis we first review the known results for the optimised Schwarz method,
deriving optimised Robin parameters and studying the asymptotic performance of the
method as the mesh parameter of the discretisation is refined and the jump in coefficients
Next we formulate the two-Lagrange multiplier method for a model two subdomain
problem and show its equivalence to the optimised Schwarz method under suitable conditions.
The two-Lagrange multiplier method results in a non-symmetric linear system which
is usually solved with a Krylov subspace method such as GMRES. The convergence of the
GMRES method can be estimated by constructing a conformal map from the exterior of
the field of values of the system matrix to the interior of the unit disc.
We approximate the field of values of the two-Lagrange multiplier system matrix by
a rectangle and calculate optimised Robin parameters that ensure the rectangle is “well
conditioned” in the sense that GMRES converges quickly. We derive convergence estimates
for GMRES and consider the behaviour asymptotically as the mesh size is refined and the
jump in coefficients becomes large.
The final part of the thesis is concerned with the case of heterogeneous problems with
many subdomains and cross points, where three or more subdomains coincide. We formulate
the two-Lagrange multiplier method for such problems and consider known preconditioners
that are needed to improve convergence as the number of subdomains increases.
Throughout the thesis numerical experiments are performed to verify the theoretical
|dc.description.sponsorship||Engineering and Physical Sciences Research Council (EPSRC) grant.||en_US
|dc.publisher||Mathematical and Computer Sciences||en_US
|dc.rights||All items in ROS are protected by the Creative Commons copyright license (http://creativecommons.org/licenses/by-nc-nd/2.5/scotland/), with some rights reserved.||
|dc.title||The optimised Schwarz method and the two-Lagrange multiplier method for heterogeneous problems||en_US