16 research outputs found
Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
In this paper, we develop a new tensor-product based preconditioner for
discontinuous Galerkin methods with polynomial degrees higher than those
typically employed. This preconditioner uses an automatic, purely algebraic
method to approximate the exact block Jacobi preconditioner by Kronecker
products of several small, one-dimensional matrices. Traditional matrix-based
preconditioners require storage and
computational work, where is the degree of basis polynomials used, and
is the spatial dimension. Our SVD-based tensor-product preconditioner requires
storage, work in two spatial
dimensions, and work in three spatial dimensions.
Combined with a matrix-free Newton-Krylov solver, these preconditioners allow
for the solution of DG systems in linear time in per degree of freedom in
2D, and reduce the computational complexity from to
in 3D. Numerical results are shown in 2D and 3D for the
advection and Euler equations, using polynomials of degree up to . For
many test cases, the preconditioner results in similar iteration counts when
compared with the exact block Jacobi preconditioner, and performance is
significantly improved for high polynomial degrees .Comment: 40 pages, 15 figure
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
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