958 research outputs found
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
Test of variational approximation for quantum chain by Monte Carlo simulation
We report results of a Monte Carlo simulation of the quantum chain.
In order to enhance the efficiency of the simulation we combine multigrid
simulation techniques with a refined discretization scheme. The resulting
accuracy of our data allows for a significant test of an analytical
approximation based on a variational ansatz. While the variational
approximation is well reproduced for a large range of parameters we find
significant deviations for low temperatures and large couplings.Comment: 12 pp. Latex + 3 figures as uuencoded compressed tar file, accepted
for publication in Phys. Lett.
An adaptive, fully implicit multigrid phase-field model for the quantitative simulation of non-isothermal binary alloy solidification
Using state-of-the-art numerical techniques, such as mesh adaptivity, implicit time-stepping and a non-linear multi-grid solver, the phase-field equations for the non-isothermal solidification of a dilute binary alloy have been solved. Using the quantitative, thin-interface formulation of the problem we have found that at high Lewis number a minimum in the dendrite tip radius is predicted with increasing undercooling, as predicted by marginal stability theory. Over the dimensionless undercooling range 0.2–0.8 the radius selection parameter, σ*, was observed to vary by over a factor of 2 and in a non-monotonic fashion, despite the anisotropy strength being constant
Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems
In this paper we present a rigorous cost and error analysis of a multilevel
estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for
lognormal diffusion problems. These problems are motivated by uncertainty
quantification problems in subsurface flow. We extend the convergence analysis
in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite
element discretizations and give a constructive proof of the
dimension-independent convergence of the QMC rules. More precisely, we provide
suitable parameters for the construction of such rules that yield the required
variance reduction for the multilevel scheme to achieve an -error
with a cost of with , and in
practice even , for sufficiently fast decaying covariance
kernels of the underlying Gaussian random field inputs. This confirms that the
computational gains due to the application of multilevel sampling methods and
the gains due to the application of QMC methods, both demonstrated in earlier
works for the same model problem, are complementary. A series of numerical
experiments confirms these gains. The results show that in practice the
multilevel QMC method consistently outperforms both the multilevel MC method
and the single-level variants even for non-smooth problems.Comment: 32 page
A dynamically adaptive multigrid algorithm for the incompressible Navier-Stokes equations: Validation and model problems
An algorithm is described for the solution of the laminar, incompressible Navier-Stokes equations. The basic algorithm is a multigrid based on a robust, box-based smoothing step. Its most important feature is the incorporation of automatic, dynamic mesh refinement. This algorithm supports generalized simple domains. The program is based on a standard staggered-grid formulation of the Navier-Stokes equations for robustness and efficiency. Special grid transfer operators were introduced at grid interfaces in the multigrid algorithm to ensure discrete mass conservation. Results are presented for three models: the driven-cavity, a backward-facing step, and a sudden expansion/contraction
- …