Scalable Parallel Multigrid Preconditioning for High Fidelity Finite Element and Finite Difference Simulations

A major challenge in undertaking high resolution numerical simulations for engineering problems comes from the growth in the computational work that occurs as the underlying finite difference or the finite element meshes are refined in order to improve accuracy. For most solution algorithms this growth in work is super-linear with the number of degrees of freedom. In such cases it is impossible for a parallel implementation with p processors to solve a problem with p × N degrees of freedom in the same time as it can solve the problem with N degrees of freedom on a single processor (so-called weak scalability). Because multigrid algorithms have the property that they may solve a discrete problem in O(N) operations they provide a natural approach for seeking weakly scalable parallel solvers. Unfortunately, developing a highly efficient parallel implementation is a challenging task, that can prevent perfect weak scalability. In this paper we explain why this is so, and suggest ways in which these difficulties may be overcome. In particular we demonstrate that, if multigrid is used as a preconditioner for a different iterative scheme (based upon Krylov subspace methods for example), the coarse grid part of the multigrid problem does not need to be solved exactly (unlike for pure multigrid) in order to obtain an O(N) algorithm. Since this is the part of multigrid that is most challenging to implement in parallel we are able to show the effectiveness of this approach for both finite difference and finite element discretizations of selected applications in both two and three dimensions.