A simple multigrid scheme for solving the Poisson equation with arbitrary domain boundaries

We present a new multigrid scheme for solving the Poisson equation with Dirichlet boundary conditions on a Cartesian grid with irregular domain boundaries. This scheme was developed in the context of the Adaptive Mesh Refinement (AMR) schemes based on a graded-octree data structure. The Poisson equation is solved on a level-by-level basis, using a "one-way interface" scheme in which boundary conditions are interpolated from the previous coarser level solution. Such a scheme is particularly well suited for self-gravitating astrophysical flows requiring an adaptive time stepping strategy. By constructing a multigrid hierarchy covering the active cells of each AMR level, we have designed a memory-efficient algorithm that can benefit fully from the multigrid acceleration. We present a simple method for capturing the boundary conditions across the multigrid hierarchy, based on a second-order accurate reconstruction of the boundaries of the multigrid levels. In case of very complex boundaries, small scale features become smaller than the discretization cell size of coarse multigrid levels and convergence problems arise. We propose a simple solution to address these issues. Using our scheme, the convergence rate usually depends on the grid size for complex grids, but good linear convergence is maintained. The proposed method was successfully implemented on distributed memory architectures in the RAMSES code, for which we present and discuss convergence and accuracy properties as well as timing performances.Comment: 33 pages, 15 figures, accepted for publication in Journal of Computational Physic

A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM

On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics

The fixed-stress split method has been widely used as solution method in the coupling of flow and geomechanics. In this work, we analyze the behavior of an inexact version of this algorithm as smoother within a geometric multigrid method, in order to obtain an efficient monolithic solver for Biot''s problem. This solver combines the advantages of being a fully coupled method, with the benefit of decoupling the flow and the mechanics part in the smoothing algorithm. Moreover, the fixed-stress split smoother is based on the physics of the problem, and therefore all parameters involved in the relaxation are based on the physical properties of the medium and are given a priori. A local Fourier analysis is applied to study the convergence of the multigrid method and to support the good convergence results obtained. The proposed geometric multigrid algorithm is used to solve several tests in semi-structured triangular grids, in order to show the good behavior of the method and its practical utility

A Parallel Geometric Multigrid Method for Adaptive Finite Elements

Applications in a variety of scientific disciplines use systems of Partial Differential Equations (PDEs) to model physical phenomena. Numerical solutions to these models are often found using the Finite Element Method (FEM), where the problem is discretized and the solution of a large linear system is required, containing millions or even billions of unknowns. Often times, the domain of these solves will contain localized features that require very high resolution of the underlying finite element mesh to accurately solve, while a mesh with uniform resolution would require far too much computational time and memory overhead to be feasible on a modern machine. Therefore, techniques like adaptive mesh refinement, where one increases the resolution of the mesh only where it is necessary, must be used. Even with adaptive mesh refinement, these systems can still be on the order of much more than a million unknowns (large mantle convection applications like the ones in [90] show simulations on over 600 billion unknowns), and attempting to solve on a single processing unit is infeasible due to limited computational time and memory required. For this reason, any application code aimed at solving large problems must be built using a parallel framework, allowing the concurrent use of multiple processing units to solve a single problem, and the code must exhibit efficient scaling to large amounts of processing units. Multigrid methods are currently the only known optimal solvers for linear systems arising from discretizations of elliptic boundary valued problems. These methods can be represented as an iterative scheme with contraction number less than one, independent of the resolution of the discretization [24, 54, 25, 103], with optimal complexity in the number of unknowns in the system [29]. Geometric multigrid (GMG) methods, where the hierarchy of spaces are defined by linear systems of finite element discretizations on meshes of decreasing resolution, have been shown to be robust for many different problem formulations, giving mesh independent convergence for highly adaptive meshes [26, 61, 83, 18], but these methods require specific implementations for each type of equation, boundary condition, mesh, etc., required by the specific application. The implementation in a massively parallel environment is not obvious, and research into this topic is far from exhaustive. We present an implementation of a massively parallel, adaptive geometric multigrid (GMG) method used in the open-source finite element library deal.II [5], and perform extensive tests showing scaling of the v-cycle application on systems with up to 137 billion unknowns run on up to 65,536 processors, and demonstrating low communication overhead of the algorithms proposed. We then show the flexibility of the GMG by applying the method to four different PDE systems: the Poisson equation, linear elasticity, advection-diffusion, and the Stokes equations. For the Stokes equations, we implement a fully matrix-free, adaptive, GMG-based solver in the mantle convection code ASPECT [13], and give a comparison to the current matrix-based method used. We show improvements in robustness, parallel scaling, and memory consumption for simulations with up to 27 billion unknowns and 114,688 processors. Finally, we test the performance of IDR(s) methods compared to the FGMRES method currently used in ASPECT, showing the effects of the flexible preconditioning used for the Stokes solves in ASPECT, and the demonstrating the possible reduction in memory consumption for IDR(s) and the potential for solving large scale problems. Parts of the work in this thesis has been submitted to peer reviewed journals in the form of two publications ([36] and [34]), and the implementations discussed have been integrated into two open-source codes, deal.II and ASPECT. From the contributions to deal.II, including a full length tutorial program, Step-63 [35], the author is listed as a contributing author to the newest deal.II release (see [5]). The implementation into ASPECT is based on work from the author and Timo Heister. The goal for the work here is to enable the community of geoscientists using ASPECT to solve larger problems than currently possible. Over the course of this thesis, the author was partially funded by the NSF Award OAC-1835452 and by the Computational Infrastructure in Geodynamics initiative (CIG), through the NSF under Award EAR-0949446 and EAR-1550901 and The University of California -- Davis

Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs

We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hierarchy is then used in a K-cycle iteration solve phase with a $\ell^1$-Jacobi smoother. Numerical tests of a parallel implementation of the method for graphics processors are presented to demonstrate its effectiveness.Comment: 18 pages, 3 figure