233 research outputs found
An assessment of solvers for saddle point problems emerging from the incompressible Navier--Stokes equations
Efficient incompressible flow simulations, using inf-sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf-sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps
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Final report on the Copper Mountain conference on multigrid methods
The Copper Mountain Conference on Multigrid Methods was held on April 6-11, 1997. It took the same format used in the previous Copper Mountain Conferences on Multigrid Method conferences. Over 87 mathematicians from all over the world attended the meeting. 56 half-hour talks on current research topics were presented. Talks with similar content were organized into sessions. Session topics included: fluids; domain decomposition; iterative methods; basics; adaptive methods; non-linear filtering; CFD; applications; transport; algebraic solvers; supercomputing; and student paper winners
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
Non-nested multi-grid solvers for mixed divergence-free Scott-Vogelius discretizations
Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure
Robust preconditioners for a new stabilized discretization of the poroelastic equations
In this paper, we present block preconditioners for a stabilized
discretization of the poroelastic equations developed in [45]. The
discretization is proved to be well-posed with respect to the physical and
discretization parameters, and thus provides a framework to develop
preconditioners that are robust with respect to such parameters as well. We
construct both norm-equivalent (diagonal) and field-of-value-equivalent
(triangular) preconditioners for both the stabilized discretization and a
perturbation of the stabilized discretization that leads to a smaller overall
problem after static condensation. Numerical tests for both two- and
three-dimensional problems confirm the robustness of the block preconditioners
with respect to the physical and discretization parameters
Schnelle Löser für partielle Differentialgleichungen
[no abstract available
Seventh Copper Mountain Conference on Multigrid Methods
The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth
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