Special Seminar: Scott MacLachlan, Tufts University

Abstract:

 

While the combination of standard finite-element discretizations and fast multigrid-based solvers is now well-established and well-understood for uniformly elliptic systems of PDEs, much work remains for problems that fall outside this class. This talk focuses on recent developments in the theory and practice of multigrid solvers for two important classes of non-uniformly elliptic systems: singular-perturbation and saddle-point problems. For singular-perturbation problems, boundary and interior layers are generated as the perturbation parameter goes to zero, leading to the need for strongly locally refined meshes to efficiently resolve the solutions to these systems. I will show how layer-aware multigrid methods can be developed to achieve optimal solver efficiency for these meshes. For saddle-point problems, difficulties arise due to the indefiniteness of the discretized linear systems, leading to many approaches based on block factorization treatment of two smaller but definite matrices. I will discuss an alternative monolithic approach, based on the constrained optimization viewpoint, to developing optimal multigrid solvers for these problems.