702 research outputs found
Multigrid Preconditioning for a Space-Time Spectral-Element Discontinuous-Galerkin Solver
In this work we examine a multigrid preconditioning approach in the context of a high- order tensor-product discontinuous-Galerkin spectral-element solver. We couple multigrid ideas together with memory lean and efficient tensor-product preconditioned matrix-free smoothers. Block ILU(0)-preconditioned GMRES smoothers are employed on the coarsest spaces. The performance is evaluated on nonlinear problems arising from unsteady scale- resolving solutions of the Navier-Stokes equations: separated low-Mach unsteady ow over an airfoil from laminar to turbulent ow. A reduction in the number of ne space iterations is observed, which proves the efficiency of the approach in terms of preconditioning the linear systems, however this gain was not reflected in the CPU time. Finally, the preconditioner is successfully applied to problems characterized by stiff source terms such as the set of RANS equations, where the simple tensor product preconditioner fails. Theoretical justification about the findings is reported and future work is outlined
Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media
In this paper we propose and analyze a preconditioner for a system arising
from a finite element approximation of second order elliptic problems
describing processes in highly het- erogeneous media. Our approach uses the
technique of multilevel methods and the recently proposed preconditioner based
on additive Schur complement approximation by J. Kraus (see [8]). The main
results are the design and a theoretical and numerical justification of an
iterative method for such problems that is robust with respect to the contrast
of the media, defined as the ratio between the maximum and minimum values of
the coefficient (related to the permeability/conductivity).Comment: 28 page
Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations
A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization
Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential
We present and analyze finite difference numerical schemes for the Allen
Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential.
Both the first order and second order accurate temporal algorithms are
considered. In the first order scheme, we treat the nonlinear logarithmic terms
and the surface diffusion term implicitly, and update the linear expansive term
and the mobility explicitly. We provide a theoretical justification that, this
numerical algorithm has a unique solution such that the positivity is always
preserved for the logarithmic arguments. In particular, our analysis reveals a
subtle fact: the singular nature of the logarithmic term around the values of
and 1 prevents the numerical solution reaching these singular values, so
that the numerical scheme is always well-defined as long as the numerical
solution stays similarly bounded at the previous time step. Furthermore, an
unconditional energy stability of the numerical scheme is derived, without any
restriction for the time step size. The unique solvability and the
positivity-preserving property for the second order scheme are proved using
similar ideas, in which the singular nature of the logarithmic term plays an
essential role. For both the first and second order accurate schemes, we are
able to derive an optimal rate convergence analysis, which gives the full order
error estimate. The case with a non-constant mobility is analyzed as well. We
also describe a practical and efficient multigrid solver for the proposed
numerical schemes, and present some numerical results, which demonstrate the
robustness of the numerical schemes
Nonconforming mortar element methods: Application to spectral discretizations
Spectral element methods are p-type weighted residual techniques for partial differential equations that combine the generality of finite element methods with the accuracy of spectral methods. Presented here is a new nonconforming discretization which greatly improves the flexibility of the spectral element approach as regards automatic mesh generation and non-propagating local mesh refinement. The method is based on the introduction of an auxiliary mortar trace space, and constitutes a new approach to discretization-driven domain decomposition characterized by a clean decoupling of the local, structure-preserving residual evaluations and the transmission of boundary and continuity conditions. The flexibility of the mortar method is illustrated by several nonconforming adaptive Navier-Stokes calculations in complex geometry
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
Spectrally approximating large graphs with smaller graphs
How does coarsening affect the spectrum of a general graph? We provide
conditions such that the principal eigenvalues and eigenspaces of a coarsened
and original graph Laplacian matrices are close. The achieved approximation is
shown to depend on standard graph-theoretic properties, such as the degree and
eigenvalue distributions, as well as on the ratio between the coarsened and
actual graph sizes. Our results carry implications for learning methods that
utilize coarsening. For the particular case of spectral clustering, they imply
that coarse eigenvectors can be used to derive good quality assignments even
without refinement---this phenomenon was previously observed, but lacked formal
justification.Comment: 22 pages, 10 figure
- …