Improved Condition Number for Spectral Methods

Heinrichs, Wilhelm

doi:10.2307/2008351

Cited by 18 publications

(30 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The iteration (19) can be combined with multigrid, in which case it serves the role of a smoother, or accelerated by Krylov-based projection methods such as conjugate gradients (if M is SPD) or GMRES [36]. One also can use multigrid as the preconditioner for conjugate gradients (CG) or GMRES.…”

Section: Iterative Solution Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

2005

View full text Add to dashboard Cite

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.

show abstract

Section: Iterative Solution Methodsmentioning

confidence: 99%

“…Figure 3 illustrates the error behavior for three variants of the hybrid Schwarz scheme for a 2 × 2 array of elements with N = 16. Figure 3(a) shows the error after a single application of the smoothing step (19) with the (additive) Schwarz preconditioner (28) with minimal overlap as depicted in Fig. 2(b).…”

Section: Richardson Lcs Gmres Lcs/gmresmentioning

confidence: 99%

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

2005

View full text Add to dashboard Cite

show abstract

“…This leads in general to full matrices, which is numerically inefficient. The modification of the standard Chebyshev-Galerkin method is basically a combination of ideas posed by Heinrichs, [11,12], Shen, [29], and also Pop, [23]. Key feature in Galerkin methods is the use of shape and test functions that satisfy a priori the boundary conditions.…”

Section: A Appendix : a Modified Chebyshev-galerkin Methodsmentioning

confidence: 99%

“…The result is projected on a finite dimensional space in order to get a finite algebraic system. Let T k (ζ ) denote the Chebyshev polynomial of degree k. We define the shape functions ( [11,12])…”

Section: A Appendix : a Modified Chebyshev-galerkin Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Transient growth in linearly stable gravity-driven flow in porous media

Pieters

Duijn

2006

European Journal of Mechanics - B/Fluids

View full text Add to dashboard Cite

In this paper we study gravitational instability of a saline boundary layer formed by evaporation induced upward throughflow at the horizontal surface of a porous medium. Van Duijn et al.,[33], derived stability bounds by means of linear stability analysis and an (improved) energy method. These bounds do not coincide, i.e. there exists a subcritical region or stability gap in the system parameter space which is due to the asymmetry of the linear part of the perturbation equations. We show that the linear operator can be symmetrized by means of a similarity transformation. For system parameter values in the stability gap, we show that there exist optimal initial perturbations for which the linearly stable system exhibits transient growth. We show that transient growth is norm dependent by considering weighted norms, which are induced by a one-parameter family of similarity transformations.

show abstract

On the accuracy of the calculation of transient growth in plane Poiseuille flow

Zhao

Duncan

2014

Numerical Methods in Fluids

View full text Add to dashboard Cite

SUMMARYThis paper addresses the accuracy of numerical methods to compute the transient energy growth of plane Poiseuille flow. We show that using the Chebyshev collocation method to discretize the linearized governing equations in the wall‐normal direction can introduce numerical problems when computing the energy evolution of the flow. We demonstrate that spurious eigenmodes of the discretized linear operator and numerical integration errors are the possible sources of the numerical problems, and we also show that spurious eigenvalues with negative real parts of large magnitude can affect the calculation of energy growth. These difficulties can be avoided by using a spectral Galerkin method where the basis functions satisfy the boundary conditions. Copyright © 2014 John Wiley & Sons, Ltd.

show abstract

Improved Condition Number for Spectral Methods

Cited by 18 publications

References 9 publications

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

Transient growth in linearly stable gravity-driven flow in porous media

On the accuracy of the calculation of transient growth in plane Poiseuille flow

Contact Info

Product

Resources

About