Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

Ingerman, David; Druskin, Vladimir; Knizhnerman, Leonid

doi:10.1002/1097-0312(200008)53:8<1039::aid-cpa4>3.3.co;2-9

Cited by 34 publications

(103 citation statements)

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“…Another approach is to use a grid which is optimal for constant coefficients for problems with variable coefficients. Such an approach would not produce spectral convergence in a strict sense, but it yields spectral convergence for singular components of the solution [14] which are the most difficult for conventional finite-differences. Thus, at no cost this grid optimization approach can be implemented for quite general linear partial differential equations (PDEs).…”

Section: Introductionmentioning

confidence: 58%

“…The identities (11), (14) and (15) combine to give us Lemma 3. Let u be the solution to (1) and u L be the Galerkin solution to (13).…”

Section: Galerkin Formulationmentioning

confidence: 99%

“…in the sense of some weighted L ∞ norm on (1, ∞) (see [14] for details). That is, we would like to find the grid which is optimal in this sense.…”

Section: Applications To Pdesmentioning

confidence: 99%

“…It is based on precomputed tables of the grids obtained for problems with constant coefficients. In some cases the tabulation is not even needed because the optimal grid can be computed via a simple analytic formula [14]. Then, as was suggested in [2,9] for wave and elliptic problems with piecewise constant coefficients and rectangular interfaces, the tables are used for the grid generation within every homogeneous subdomain.…”

Section: Introductionmentioning

confidence: 99%

“…First we show how one can use one-dimensional optimal grids to greatly improve efficiency for Laplace's equation on a strip and a one-dimensional wave equation. Some of these and other examples have been demonstrated numerically in [2,14]. Next we prove that by taking the tensor product of a one-dimensional optimal grid with itself, one can obtain a two-dimensional five-point scheme with convergence equivalent to a two dimensional spectral Galerkin method at one corner.…”

Section: Introductionmentioning

confidence: 99%

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Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation

Druskin

Moskow

2001

Math. Comp.

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Abstract. A method for calculating special grid placement for three-point schemes which yields exponential superconvergence of the Neumann to Dirichlet map has been suggested earlier. Here we show that such a grid placement can yield impedance which is equivalent to that of a spectral Galerkin method, or more generally to that of a spectral Galerkin-Petrov method. In fact we show that for every stable Galerkin-Petrov method there is a threepoint scheme which yields the same solution at the boundary. We discuss the application of this result to partial differential equations and give numerical examples. We also show equivalence at one corner of a two-dimensional optimal grid with a spectral Galerkin method.

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Section: Introductionmentioning

confidence: 58%

“…The identities (11), (14) and (15) combine to give us Lemma 3. Let u be the solution to (1) and u L be the Galerkin solution to (13).…”

Section: Galerkin Formulationmentioning

confidence: 99%

“…in the sense of some weighted L ∞ norm on (1, ∞) (see [14] for details). That is, we would like to find the grid which is optimal in this sense.…”

Section: Applications To Pdesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation

Druskin

Moskow

2001

Math. Comp.

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A GPU‐accelerated nodal discontinuous Galerkin method with high‐order absorbing boundary conditions and corner/edge compatibility

Modave

Atle²,

Chan

et al. 2017

Numerical Meth Engineering

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Discontinuous Galerkin finite element schemes exhibit attractive features for accurate large-scale wave-propagation simulations on modern parallel architectures. For many applications, these schemes must be coupled with non-reflective boundary treatments to limit the size of the computational domain without losing accuracy or computational efficiency, which remains a challenging task. In this paper, we present a combination of a nodal discontinuous Galerkin method with high-order absorbing boundary conditions (HABCs) for cuboidal computational domains. Compatibility conditions are derived for HABCs intersecting at the edges and the corners of a cuboidal domain. We propose a GPU implementation of the computational procedure, which results in a multidimensional solver with equations to be solved on 0D, 1D, 2D and 3D spatial regions. Numerical results demonstrate both the accuracy and the computational efficiency of our approach.

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Numerical methods for the QCD overlap operator IV: Hybrid Monte Carlo

Cundy

Krieg

Arnold

et al. 2009

Computer Physics Communications

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The computational costs of calculating the matrix sign function of the overlap operator together with fundamental numerical problems related to the discontinuity of the sign function in the kernel eigenvalues are the major obstacles towards simulations with dynamical overlap fermions using the Hybrid Monte Carlo algorithm. In a previous paper of the present series we introduced optimal numerical approximation of the sign function and have developed highly advanced preconditioning and relaxation techniques which speed up the the inversion of the overlap operator by nearly an order of magnitude.In this fourth paper of the series we construct an HMC algorithm for overlap fermions. We approximate the matrix sign function using the Zolotarev rational approximation, treating the smallest eigenvalues of the Wilson operator exactly within the fermionic force. Based on this we derive the fermionic force for the overlap operator. We explicitly solve the problem of the Dirac delta-function terms arising through zero crossings of eigenvalues of the Wilson operator. The main advantage of this scheme is that its energy violations scale better than O(∆τ 2 ) and thus are comparable with the violations of the standard leapfrog algorithm over the course of a trajectory. We explicitly prove that our algorithm satisfies reversibility and area conservation. We present test results from our algorithm on 4 4 , 6 4 , and 8 4 lattices.

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Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

Cited by 34 publications

References 0 publications

Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation

Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation

A GPU‐accelerated nodal discontinuous Galerkin method with high‐order absorbing boundary conditions and corner/edge compatibility

Numerical methods for the QCD overlap operator IV: Hybrid Monte Carlo

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