576 research outputs found
The cost of continuity: performance of iterative solvers on isogeometric finite elements
In this paper we study how the use of a more continuous set of basis
functions affects the cost of solving systems of linear equations resulting
from a discretized Galerkin weak form. Specifically, we compare performance of
linear solvers when discretizing using B-splines, which span traditional
finite element spaces, and B-splines, which represent maximum
continuity. We provide theoretical estimates for the increase in cost of the
matrix-vector product as well as for the construction and application of
black-box preconditioners. We accompany these estimates with numerical results
and study their sensitivity to various grid parameters such as element size
and polynomial order of approximation . Finally, we present timing results
for a range of preconditioning options for the Laplace problem. We conclude
that the matrix-vector product operation is at most \slfrac{33p^2}{8} times
more expensive for the more continuous space, although for moderately low ,
this number is significantly reduced. Moreover, if static condensation is not
employed, this number further reduces to at most a value of 8, even for high
. Preconditioning options can be up to times more expensive to setup,
although this difference significantly decreases for some popular
preconditioners such as Incomplete LU factorization
Recommended from our members
Preparing sparse solvers for exascale computing.
Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'
Schwarz type preconditioners for the neutron diffusion equation
[EN] Domain decomposition is a mature methodology that has been used to accelerate the convergence of partial differential equations. Even if it was devised as a solver by itself, it is usually employed together with Krylov iterative methods improving its rate of convergence, and providing scalability with respect to the size of the problem.
In this work, a high order finite element discretization of the neutron diffusion equation is considered. In this problem the preconditioning of large and sparse linear systems arising from a source driven formulation becomes necessary due to the complexity of the problem. On the other hand, preconditioners based on an incomplete factorization are very expensive from the point of view of memory requirements. The acceleration of the neutron diffusion equation is thus studied here by using alternative preconditioners based on domain decomposition techniques inside Schur complement methodology. The study considers substructuring preconditioners, which do not involve overlapping, and additive Schwarz preconditioners, where some overlapping between the subdomains is taken into account.
The performance of the different approaches is studied numerically using two-dimensional and three-dimensional problems. It is shown that some of the proposed methodologies outperform incomplete LU factorization for preconditioning as long as the linear system to be solved is large enough, as it occurs for three-dimensional problems. They also outperform classical diagonal Jacobi preconditioners, as long as the number of systems to be solved is large enough in such a way that the overhead of building the pre-conditioner is less than the improvement in the convergence rate. (C) 2016 Elsevier B.V. All rights reserved.The work has been partially supported by the spanish Ministerio de Economía y Competitividad under projects ENE 2014-59442-P and MTM2014-58159-P, the Generalitat Valenciana under the project PROMETEO II/2014/008 and the Universitat Politècnica de València under the project FPI-2013. The work has also been supported partially by the Swedish Research Council (VR-Vetenskapsrådet) within a framework grant called DREAM4SAFER, research contract C0467701.Vidal-Ferràndiz, A.; González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ.; Demazière, C. (2017). Schwarz type preconditioners for the neutron diffusion equation. Journal of Computational and Applied Mathematics. 309:563-574. https://doi.org/10.1016/j.cam.2016.02.056S56357430
A Numerical Approach to Space-Time Finite Elements for the Wave Equation
We study a space-time finite element approach for the nonhomogeneous wave
equation using a continuous time Galerkin method. We present fully implicit
examples in 1+1, 2+1, and 3+1 dimensions using linear quadrilateral,
hexahedral, and tesseractic elements. Krylov solvers with additive Schwarz
preconditioning are used for solving the linear system. We introduce a time
decomposition strategy in preconditioning which significantly improves
performance when compared with unpreconditioned cases.Comment: 9 pages, 5 figures, 5 table
Uniform subspace correction preconditioners for discontinuous Galerkin methods with -refinement
In this paper, we develop subspace correction preconditioners for
discontinuous Galerkin (DG) discretizations of elliptic problems with
-refinement. These preconditioners are based on the decomposition of the DG
finite element space into a conforming subspace, and a set of small
nonconforming edge spaces. The conforming subspace is preconditioned using a
matrix-free low-order refined technique, which in this work we extend to the
-refinement context using a variational restriction approach. The condition
number of the resulting linear system is independent of the granularity of the
mesh , and the degree of polynomial approximation . The method is
amenable to use with meshes of any degree of irregularity and arbitrary
distribution of polynomial degrees. Numerical examples are shown on several
test cases involving adaptively and randomly refined meshes, using both the
symmetric interior penalty method and the second method of Bassi and Rebay
(BR2).Comment: 24 pages, 9 figure
- …