The research in this project is on the theory and applications of multigrid methods. One of the goals is to generalize the investigator's additive multigrid theory, which can handle the convergence of V-cycle and F-cycle algorithms for nonconforming methods, to more difficult problems such as anisotropic problems, nonsymmetric problems and indefinite problems, and to new discretization techniques such as mortar finite element methods and discontinuous Galerkin methods. Another goal is to extend the PI's multigrid method for singular solutions and stress intensity factors to more complicated problems and to three dimensions. This new multigrid approach can recover the optimal convergence rates of simple finite elements on simple grids, even in the presence of strong singularities caused by nonsmooth geometries, abrupt changes in boundary conditions, or jumps in the coefficients of partial differential equations. It can also take full advantage of superconvergence phenomena, extrapolation techniques, and parallel implementations.
Multigrid methods can produce fast solutions to large systems of equations. The errors of multigrid solutions are comparable to the smallest possible errors and at the same time the computational cost of multigrid methods is proportional to the number of unknowns. Therefore multigrid methods have optimal complexity, and they (either on their own or combined with other methods) are powerful engines for large scale scientific computations. The results of this project can provide answers to the important question of the reliability of multigrid methods and provide guidelines for the development of new algorithms. They will also generate useful computational tools for many challenging problems in material science, fracture mechanics, fluid flow and electromagnetism.
