469 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
Analysis of a parallel multigrid algorithm
The parallel multigrid algorithm of Frederickson and McBryan (1987) is considered. This algorithm uses multiple coarse-grid problems (instead of one problem) in the hope of accelerating convergence and is found to have a close relationship to traditional multigrid methods. Specifically, the parallel coarse-grid correction operator is identical to a traditional multigrid coarse-grid correction operator, except that the mixing of high and low frequencies caused by aliasing error is removed. Appropriate relaxation operators can be chosen to take advantage of this property. Comparisons between the standard multigrid and the new method are made
A local Fourier analysis of additive Vanka relaxation for the Stokes equations
Multigrid methods are popular solution algorithms for many discretized PDEs,
either as standalone iterative solvers or as preconditioners, due to their high
efficiency. However, the choice and optimization of multigrid components such
as relaxation schemes and grid-transfer operators is crucial to the design of
optimally efficient algorithms. It is well--known that local Fourier analysis
(LFA) is a useful tool to predict and analyze the performance of these
components. In this paper, we develop a local Fourier analysis of monolithic
multigrid methods based on additive Vanka relaxation schemes for mixed
finite-element discretizations of the Stokes equations. The analysis offers
insight into the choice of "patches" for the Vanka relaxation, revealing that
smaller patches offer more effective convergence per floating point operation.
Parameters that minimize the two-grid convergence factor are proposed and
numerical experiments are presented to validate the LFA predictions.Comment: 30 pages, 12 figures. Add new sections: multiplicative Vanka results
and sensitivity of convergence factors to mesh distortio
Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations
We analyse two practical aspects that arise in the numerical solution of
Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone
approximation schemes known as semi-Lagrangian schemes. These schemes make use
of a wide stencil to achieve convergence and result in discretization matrices
that are less sparse and less local than those coming from standard finite
difference schemes. This leads to computational difficulties not encountered
there. In particular, we consider the overstepping of the domain boundary and
analyse the accuracy and stability of stencil truncation. This truncation
imposes a stricter CFL condition for explicit schemes in the vicinity of
boundaries than in the interior, such that implicit schemes become attractive.
We then study the use of geometric, algebraic and aggregation-based multigrid
preconditioners to solve the resulting discretised systems from implicit time
stepping schemes efficiently. Finally, we illustrate the performance of these
techniques numerically for benchmark test cases from the literature
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