Algebraic Multigrid with Optimal Interpolation and Adaptive Smoothers (Final Report)

Abstract

The project team continued with work on developing new bootstrap AMG techniques for solving symmetric and non-symmetric PDE systems. The focus of this work is to derive more reliable measures of the quality of the coarse space set than the convergence rate of the standard F-relaxation form of CR and a more robust form of interpolation than the so-called ideal form. We have successfully derived a sharp variant of CR that gives the precise convergence rate of the two-grid method using this optimal interpolation and, in addition, we derived a new Generalized Bootstrap AMG setup algorithm that uses as its main tool a multilevel eigensolver for the generalized eigenvalue problem involving the system matrix and the symmetrized smoother. In addition, the approach allows for general block smoothers with overlap. We have applied the method to scalar diffusion problems, linear elasticity, and Maxwell’s and the method shows marked improvements over existing AMG methods for these problems. In addition, the team worked with CASC members on new forms of ideal AMG interpolation.