Algebraic Multigrid Methods and Their Application to Generalized Finite Element Methods

The research in this proposal is on the study and applications of

efficient algebraic multigrid methods for the solution of linear

algebraic systems arising from the discretization of second order

partial differential equations by the generalized finite element

method. The proposed research will focus on the development and

analysis of adaptive techniques in the construction of hierarchy of

nested spaces and the choice of approximate subspace solvers that lead

to the efficient and robust multigrid methods applicable to wide range

of generalized finite element discretizations.

The rapid increase in the power of today's supercomputers has made it

feasible for the scientific community to use numerical simulations to

model physical phenomena to produce meaningful results. One of the

modern techniques that can deliver quantitative results via such

simulations is the generalized finite element method. This method has

proved to be a very robust discretization tool, applicable in various

branches of engineering and sciences, for example, in simulating and

determining the elastic, electromagnetic and other important physical

properties of heterogeneous materials. Like most other discretization

techniques, most often the majority of computation in such simulations

is devoted to the solution of the resulting linear systems of

equations. Hence, it is very important to develop efficient solvers

for these systems. The results from the proposed research are thus

expected to have a broad and noticeable impact by providing the much

needed iterative multilevel solution techniques for the discrete

linear systems arising from numerical models in many applications.

The proposed research is also expected to have an educational impact

as it will provide a solid base for training of graduate students in

the modern theoretical and practical aspects of numerical methods for

problems in science and engineering.