Hybrid multigrid methods for high-order discontinuous Galerkin discretizations

The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE solver. The multigrid hierarchy exploits all possibilities of geometric, polynomial, and algebraic coarsening, targeting engineering applications on complex geometries. Additionally, a transfer from discontinuous to continuous function spaces is performed within the multigrid hierarchy. This does not only further reduce the problem size of the coarse-grid problem, but also leads to a discretization most suitable for state-of-the-art algebraic multigrid methods applied as coarse-grid solver. The relevant design choices regarding the selection of optimal multigrid coarsening strategies among the various possibilities are discussed with the metric of computational costs as the driving force for algorithmic selections. We find that a transfer to a continuous function space at highest polynomial degree (or on the finest mesh), followed by polynomial and geometric coarsening, shows the best overall performance. The success of this particular multigrid strategy is due to a significant reduction in iteration counts as compared to a transfer from discontinuous to continuous function spaces at lowest polynomial degree (or on the coarsest mesh). The coarsening strategy with transfer to a continuous function space on the finest level leads to a multigrid algorithm that is robust with respect to the penalty parameter of the SIPG method. Detailed numerical investigations are conducted for a series of examples ranging from academic test cases to more complex, practically relevant geometries. Performance comparisons to state-of-the-art methods from the literature demonstrate the versatility and computational efficiency of the proposed multigrid algorithms

The Mixed Finite Element Multigrid Method for Stokes Equations

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q2-Q1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results

Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by B\"{o}rm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the analysis to three dimensions under slightly weakened assumptions, and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment (DUNE), which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR supercomputer.Comment: 22 pages, 6 Figures, 2 Table

Non-invasive multigrid for semi-structured grids

Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to promote the efficient utilization of high performance computer architectures. These HHG meshes are constructed by uniformly refining a relatively coarse fully unstructured mesh. While HHG meshes provide some flexibility for unstructured applications, most multigrid calculations can be accomplished using efficient structured grid ideas and kernels. This paper focuses on generalizing the HHG idea so that it is applicable to a broader community of computational scientists, and so that it is easier for existing applications to leverage structured multigrid components. Specifically, we adapt the structured multigrid methodology to significantly more complex semi-structured meshes. Further, we illustrate how mature applications might adopt a semi-structured solver in a relatively non-invasive fashion. To do this, we propose a formal mathematical framework for describing the semi-structured solver. This formalism allows us to precisely define the associated multigrid method and to show its relationship to a more traditional multigrid solver. Additionally, the mathematical framework clarifies the associated software design and implementation. Numerical experiments highlight the relationship of the new solver with classical multigrid. We also demonstrate the generality and potential performance gains associated with this type of semi-structured multigrid