191 research outputs found
End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations
In this paper, we present algorithms and implementations for the end-to-end
GPU acceleration of matrix-free low-order-refined preconditioning of high-order
finite element problems. The methods described here allow for the construction
of effective preconditioners for high-order problems with optimal memory usage
and computational complexity. The preconditioners are based on the construction
of a spectrally equivalent low-order discretization on a refined mesh, which is
then amenable to, for example, algebraic multigrid preconditioning. The
constants of equivalence are independent of mesh size and polynomial degree.
For vector finite element problems in and (e.g.
for electromagnetic or radiation diffusion problems) a specially constructed
interpolation-histopolation basis is used to ensure fast convergence. Detailed
performance studies are carried out to analyze the efficiency of the GPU
algorithms. The kernel throughput of each of the main algorithmic components is
measured, and the strong and weak parallel scalability of the methods is
demonstrated. The different relative weighting and significance of the
algorithmic components on GPUs and CPUs is discussed. Results on problems
involving adaptively refined nonconforming meshes are shown, and the use of the
preconditioners on a large-scale magnetic diffusion problem using all spaces of
the finite element de Rham complex is illustrated.Comment: 23 pages, 13 figure
Parallel Element-Based Algebraic Multigrid for H (Curl) And H (Div) Problems Using the Parelag Library
This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers (ParELAG) library, to produce multilevel preconditioners and solvers for H (curl) and H (div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and the Auxiliary-Space Maxwell Solver or the Auxiliary-Space Divergence Solver on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library
Robust Preconditioners for Incompressible MHD Models
In this paper, we develop two classes of robust preconditioners for the
structure-preserving discretization of the incompressible magnetohydrodynamics
(MHD) system. By studying the well-posedness of the discrete system, we design
block preconditioners for them and carry out rigorous analysis on their
performance. We prove that such preconditioners are robust with respect to most
physical and discretization parameters. In our proof, we improve the existing
estimates of the block triangular preconditioners for saddle point problems by
removing the scaling parameters, which are usually difficult to choose in
practice. This new technique is not only applicable to the MHD system, but also
to other problems. Moreover, we prove that Krylov iterative methods with our
preconditioners preserve the divergence-free condition exactly, which
complements the structure-preserving discretization. Another feature is that we
can directly generalize this technique to other discretizations of the MHD
system. We also present preliminary numerical results to support the
theoretical results and demonstrate the robustness of the proposed
preconditioners
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