HAZniCS -- Software Components for Multiphysics Problems

We introduce the software toolbox HAZniCS for solving interface-coupled multiphysics problems. HAZniCS is a suite of modules that combines the well-known FEniCS framework for finite element discretization with solver and graph library HAZmath. The focus of the paper is on the design and implementation of a pool of robust and efficient solver algorithms which tackle issues related to the complex interfacial coupling of the physical problems often encountered in applications in brain biomechanics. The robustness and efficiency of the numerical algorithms and methods is shown in several numerical examples, namely the Darcy-Stokes equations that model flow of cerebrospinal fluid in the human brain and the mixed-dimensional model of electrodiffusion in the brain tissue

Multigrid methods for complex engineering geometries and unstructured meshes

The convergence of standard multigrid methods decays significantly if locally poor quality cells are present, and it is found that the poor convergence is due to the local failure of the smoothing property. The high frequency error localised in regions of low quality cells is not eliminated by standard multigrid smoothers and persists through multigrid cycles. We propose a global–local combined smoother for the geometric multigrid to deal with engineering meshes with a small number of poor quality cells, which includes two steps: a global smoother on the whole domain, followed by a local correction on the subdomains with low quality cells. The high frequency error remaining in the low quality regions can be damped out completely by the local correction. The idea is extended to the algebraic multigrid (AMG), including both classical AMG and smoothed aggregation AMG. It is suggested that the high frequency error produced by the smoother propagates outward the low quality region on the fine grid to the neighbouring areas on the coarse grid. An algorithm to track low quality regions on the abstract coarse grid of AMG has been developed based on the information transfer between grid levels via the transfer operators. With the local correction applied on low quality regions tracked on the abstract coarse grid, the high frequency error due to low grid quality can be removed. In the smoothed aggregation AMG, the construction of the smoothed prolongation operator depends on the spectral radius of the system. However, regions of low quality cells in a mesh increase the largest eigenvalue of the linear system. We propose a shifted largest eigenvalue strategy to approximate a reasonable spectral radius to construct the smoothed prolongation. Two and three dimensional numerical experiments, from illustrate to complicated, are demonstrated to validate the proposed smoother. Elliptic type PDEs, including Poisson and elasticity problems, are solved. For each example, the performance of multigrid on a high quality mesh is also presented as a reference case, and it is shown that the poor convergence of multigrid for low quality meshes can be recovered to the reference case by the proposed smoother. A realistic thermomechanical simulation of turbomachinery problem has also been successfully solved

Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

Optimization and coarse-grid selection for algebraic multigrid

Multigrid methods are often the most efficient approaches for solving the very large linear systems that arise from discretized PDEs and other problems. Algebraic multigrid (AMG) methods are used when the discretization lacks the structure needed to enable more efficient geometric multigrid techniques. AMG methods rely in part on heuristic graph algorithms to achieve their performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these heuristics. The main focus of this thesis is to develop e↵ective algebraic multigrid methods. A key step in all AMG approaches is the choice of the coarse/fine partitioning, aiming to balance the convergence of the iteration with its cost. In past work (MacLachlan and Saad, A greedy strategy for coarse-grid selection, SISC 2007), a constrained combinatorial optimization problem was used to define the “best” coarse grid within the setting of two-level reduction-based AMG and was shown to be NP-complete. In the first part of the thesis, a new coarsening algorithm based on simulated annealing has been developed to solve this problem. The new coarsening algorithm gives better results than the greedy algorithm developed previously. The goal of the second part of the thesis is to improve the classical AMGr method. Convergence factor bounds do not hold when AMGr algorithms are applied to matrices that are not diagonally dominant. In this part of our research, we present modifications to the classical AMGr algorithm that improve its performance on such matrices. For non-diagonally dominant matrices, we find that strength of connection plays a vital role in the performance of AMGr. To generalize the diagonal approximations of AFF used in classical AMGr, we use a sparse approximate inverse (SPAI) method, with nonzero pattern determined by strong connections, to define the AMGr-style interpolation operator, coupled with rescaling based on relaxed vectors. We present numerical results demonstrating the robustness of this approach for non-diagonally dominant systems. In the third part of this research, we have developed an improved deterministic coarsening algorithm that generalizes an existing technique known as Lloyd’s algorithm. The improved algorithm provides better control of the number of clusters than classical approaches and attempts to provide more “compact” groupings