6,027 research outputs found
A two-level algorithm for the weak Galerkin discretization of diffusion problems
This paper analyzes a two-level algorithm for the weak Galerkin (WG) finite
element methods based on local Raviart-Thomas (RT) and Brezzi-Douglas-Marini
(BDM) mixed elements for two- and three-dimensional diffusion problems with
Dirichlet condition. We first show the condition numbers of the stiffness
matrices arising from the WG methods are of . We use an extended
version of the Xu-Zikatanov (XZ) identity to derive the convergence of the
algorithm without any regularity assumption. Finally we provide some numerical
results
Low-Rank Solution Methods for Stochastic Eigenvalue Problems
We study efficient solution methods for stochastic eigenvalue problems
arising from discretization of self-adjoint partial differential equations with
random data. With the stochastic Galerkin approach, the solutions are
represented as generalized polynomial chaos expansions. A low-rank variant of
the inverse subspace iteration algorithm is presented for computing one or
several minimal eigenvalues and corresponding eigenvectors of
parameter-dependent matrices. In the algorithm, the iterates are approximated
by low-rank matrices, which leads to significant cost savings. The algorithm is
tested on two benchmark problems, a stochastic diffusion problem with some
poorly separated eigenvalues, and an operator derived from a discrete
stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup
stability constant. Numerical experiments show that the low-rank algorithm
produces accurate solutions compared to the Monte Carlo method, and it uses
much less computational time than the original algorithm without low-rank
approximation
Large-scale Optimization-based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies
It is well-known that the standard Galerkin formulation, which is often the
formulation of choice under the finite element method for solving self-adjoint
diffusion equations, does not meet maximum principles and the non-negative
constraint for anisotropic diffusion equations. Recently, optimization-based
methodologies that satisfy maximum principles and the non-negative constraint
for steady-state and transient diffusion-type equations have been proposed. To
date, these methodologies have been tested only on small-scale academic
problems. The purpose of this paper is to systematically study the performance
of the non-negative methodology in the context of high performance computing
(HPC). PETSc and TAO libraries are, respectively, used for the parallel
environment and optimization solvers. For large-scale problems, it is important
for computational scientists to understand the computational performance of
current algorithms available in these scientific libraries. The numerical
experiments are conducted on the state-of-the-art HPC systems, and a
single-core performance model is used to better characterize the efficiency of
the solvers. Our studies indicate that the proposed non-negative computational
framework for diffusion-type equations exhibits excellent strong scaling for
real-world large-scale problems
Numerical methods for nonlocal and fractional models
Partial differential equations (PDEs) are used, with huge success, to model
phenomena arising across all scientific and engineering disciplines. However,
across an equally wide swath, there exist situations in which PDE models fail
to adequately model observed phenomena or are not the best available model for
that purpose. On the other hand, in many situations, nonlocal models that
account for interaction occurring at a distance have been shown to more
faithfully and effectively model observed phenomena that involve possible
singularities and other anomalies. In this article, we consider a generic
nonlocal model, beginning with a short review of its definition, the properties
of its solution, its mathematical analysis, and specific concrete examples. We
then provide extensive discussions about numerical methods, including finite
element, finite difference, and spectral methods, for determining approximate
solutions of the nonlocal models considered. In that discussion, we pay
particular attention to a special class of nonlocal models that are the most
widely studied in the literature, namely those involving fractional
derivatives. The article ends with brief considerations of several modeling and
algorithmic extensions which serve to show the wide applicability of nonlocal
modeling.Comment: Revised/Improved version. 126 pages, 18 figures, review pape
High-performance Implementation of Matrix-free High-order Discontinuous Galerkin Methods
Achieving a substantial part of peak performance on todays and future
high-performance computing systems is a major challenge for simulation codes.
In this paper we address this question in the context of the numerical solution
of partial differential equations with finite element methods, in particular
the discontinuous Galerkin method applied to a convection-diffusion-reaction
model problem. Assuming tensor product structure of basis functions and
quadrature on cuboid meshes in a matrix-free approach a substantial reduction
in computational complexity can be achieved for operator application compared
to a matrix-based implementation while at the same time enabling SIMD
vectorization and the use of fused-multiply-add. Close to 60\% of peak
performance are obtained for a full operator evaluation on a Xeon Haswell CPU
with 16 cores and speedups of several hundred (with respect to matrix-based
computation) are achieved for polynomial degree seven. Excellent weak
scalability on a single node as well as the roofline model demonstrate that the
algorithm is fully compute-bound with a high flop per byte ratio. Excellent
scalability is also demonstrated on up to 6144 cores using message passing.Comment: submitted to SIAM SISC on 2017-11-2
Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids
In this paper two new families of arbitrary high order accurate spectral DG
finite element methods are derived on staggered Cartesian grids for the
solution of the inc.NS equations in two and three space dimensions. Pressure
and velocity are expressed in the form of piecewise polynomials along different
meshes. While the pressure is defined on the control volumes of the main grid,
the velocity components are defined on a spatially staggered mesh. In the first
family, h.o. of accuracy is achieved only in space, while a simple
semi-implicit time discretization is derived for the pressure gradient in the
momentum equation. The resulting linear system for the pressure is symmetric
and positive definite and either block 5-diagonal (2D) or block 7-diagonal (3D)
and can be solved very efficiently by means of a classical matrix-free
conjugate gradient method. The use of a preconditioner was not necessary. This
is a rather unique feature among existing implicit DG schemes for the NS
equations. In order to avoid a stability restriction due to the viscous terms,
the latter are discretized implicitly. The second family of staggered DG
schemes achieves h.o. of accuracy also in time by expressing the numerical
solution in terms of piecewise space-time polynomials. In order to circumvent
the low order of accuracy of the adopted fractional stepping, a simple
iterative Picard procedure is introduced. In this manner, the symmetry and
positive definiteness of the pressure system are not compromised. The resulting
algorithm is stable, computationally very efficient, and at the same time
arbitrary h.o. accurate in both space and time. The new numerical method has
been thoroughly validated for approximation polynomials of degree up to N=11,
using a large set of non-trivial test problems in two and three space
dimensions, for which either analytical, numerical or experimental reference
solutions exist.Comment: 46 pages, 15 figures, 4 table
A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations
In this study, we consider the numerical solution of large systems of linear
equations obtained from the stochastic Galerkin formulation of stochastic
partial differential equations. We propose an iterative algorithm that exploits
the Kronecker product structure of the linear systems. The proposed algorithm
efficiently approximates the solutions in low-rank tensor format. Using
standard Krylov subspace methods for the data in tensor format is
computationally prohibitive due to the rapid growth of tensor ranks during the
iterations. To keep tensor ranks low over the entire iteration process, we
devise a rank-reduction scheme that can be combined with the iterative
algorithm. The proposed rank-reduction scheme identifies an important subspace
in the stochastic domain and compresses tensors of high rank on-the-fly during
the iterations. The proposed reduction scheme is a multilevel method in that
the important subspace can be identified inexpensively in a coarse spatial grid
setting. The efficiency of the proposed method is illustrated by numerical
experiments on benchmark problems
On Adaptive Eulerian-Lagrangian Method for Linear Convection-Diffusion Problems
In this paper, we consider the adaptive Eulerian--Lagrangian method (ELM) for
linear convection-diffusion problems. Unlike the classical a posteriori error
estimations, we estimate the temporal error along the characteristics and
derive a new a posteriori error bound for ELM semi-discretization. With the
help of this proposed error bound, we are able to show the optimal convergence
rate of ELM for solutions with minimal regularity. Furthermore, by combining
this error bound with a standard residual-type estimator for the spatial error,
we obtain a posteriori error estimators for a fully discrete scheme. We present
numerical tests to demonstrate the efficiency and robustness of our adaptive
algorithm.Comment: 30 page
Nonsymmetric Reduction-based Algebraic Multigrid
Algebraic multigrid (AMG) is often an effective solver for symmetric positive
definite (SPD) linear systems resulting from the discretization of general
elliptic PDEs, or the spatial discretization of parabolic PDEs. However,
convergence theory and most variations of AMG rely on being SPD. Hyperbolic
PDEs, which arise often in large-scale scientific simulations, remain a
challenge for AMG, as well as other fast linear solvers, in part because the
resulting linear systems are often highly nonsymmetric. Here, a novel
convergence framework is developed for nonsymmetric, reduction-based AMG, and
sufficient conditions derived for -convergence of error and residual.
In particular, classical multigrid approximation properties are connected with
reduction-based measures to develop a robust framework for nonsymmetric,
reduction-based AMG.
Matrices with block-triangular structure are then recognized as being
amenable to reduction-type algorithms, and a reduction-based AMG method is
developed for upwind discretizations of hyperbolic PDEs, based on the concept
of a Neumann approximation to ideal restriction (AIR). AIR can be seen as
a variation of local AIR (AIR) introduced in previous work, specifically
targeting matrices with triangular structure. Although less versatile than
AIR, setup times for AIR can be substantially faster for problems with
high connectivity. AIR is shown to be an effective and scalable solver of
steady state transport for discontinuous, upwind discretizations, with
unstructured meshes, and up to 6th-order finite elements, offering a
significant improvement over existing AMG methods. AIR is also shown to be
effective on several classes of `nearly triangular' matrices, resulting from
curvilinear finite elements and artificial diffusion.Comment: Accepted SIAM Journal on Scientific Computing (Sep. 2019
On the sensitivity to model parameters in a filter stabilization technique for advection dominated advection-diffusion-reaction problems
We consider a filter stabilization technique with a deconvolution-based
indicator function for the simulation of advection dominated
advection-diffusion-reaction (ADR) problems with under-refined meshes. The
proposed technique has been previously applied to the incompressible
Navier-Stokes equations and has been successfully validated against
experimental data. However, it was found that some key parameters in this
approach have a strong impact on the solution. To better understand the role of
these parameters, we consider ADR problems, which are simpler than
incompressible flow problems. For the implementation of the filter
stabilization technique to ADR problems we adopt a three-step algorithm that
requires (i) the solution of the given problem on an under-refined mesh, (ii)
the application of a filter to the computed solution, and (iii) a relaxation
step. We compare our deconvolution-based approach to classical stabilization
methods and test its sensitivity to model parameters on a 2D benchmark problem
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