An integrated Davidson and multigrid solution approach for very large scale symmetric eigenvalue problems

Abstract

The Davidson method has recently emerged as one of the most popular solution approaches for large scale eigenvalue problems. The performance is however dominated by the choice of the correction equation and the way in which the equation is solved. In this paper, the most popular choices of correction equations are evaluated based on the three selection criteria. The Jacobi-Davidson and (modified) Inflated Newton schemes are identified as the two best candidates. A highly effective solution approach – multi-grid (MG) – is considered for the solution of the selected correction equations and the issues associated with these two equations are discussed. In addition, a two-level integrated MG and Davidson solution strategy is presented to further enhance the performance of the Davidson method. Finally, the behaviour of the two correction equations is assessed numerically over a set of large scale 3D examples with up to one million d.o.f.