Discrete Sobolev spaces and regularity of elliptic difference schemes

Stevenson, Rob

doi:10.1051/m2an/1991250506071

Cited by 8 publications

(7 citation statements)

References 10 publications

(21 reference statements)

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“…We will need to know that a grid function on the irregular points near the interface is almost the discrete divergence of a function which is smaller by a factor of h. The following is proved as Lemma 8.1 in [4]. Related statements are Lemma 2.2 in [11], Lemma 2.10 in [24], and Lemmas 2.2 and 2.6 in [3].…”

Section: Operators On Periodic Gridsmentioning

confidence: 99%

Uniform Error Estimates for Navier--Stokes Flow with an Exact Moving Boundary Using the Immersed Interface Method

Beale¹

2015

SIAM J. Numer. Anal.

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We prove that uniform accuracy of almost second order can be achieved with a finite difference method applied to Navier-Stokes flow at low Reynolds number with a moving boundary, or interface, creating jumps in the velocity gradient and pressure. Difference operators are corrected to O(h) near the interface using the immersed interface method, adding terms related to the jumps, on a regular grid with spacing h and periodic boundary conditions. The force at the interface is assumed known within an error tolerance; errors in the interface location are not taken into account. The error in velocity is shown to be uniformly O(h 2 | log h| 2 ), even at grid points near the interface, and, up to a constant, the pressure has error O(h 2 | log h| 3 ). The proof uses estimates for finite difference versions of Poisson and diffusion equations which exhibit a gain in regularity in maximum norm.

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Section: Operators On Periodic Gridsmentioning

confidence: 99%

Uniform Error Estimates for Navier--Stokes Flow with an Exact Moving Boundary Using the Immersed Interface Method

Beale¹

2015

SIAM J. Numer. Anal.

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“…In the foregoing context, the definitions stated by [29,30] are used to provide an operator theoretic formulation of (20) in the spirit of (12). Throughout this paper, the discrete spaces H s = H s ( Z) for s = + 1 2 and − 1 2 are considered.…”

Section: Wiener-hopf Formulation Of Discrete Sommerfeld Problems On Smentioning

confidence: 99%

“…In the context of the promised theme of this paper, in order to move further, an identification is required between the continuous and discrete formulation. This is accomplished by a suitable definition of prolongation and restriction operators [30]. Remark 2.9.…”

Section: Wiener-hopf Formulation Of Discrete Sommerfeld Problems On Smentioning

confidence: 99%

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Continuum limit of discrete Sommerfeld problems on square lattice

Sharma

2017

Sādhanā

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A low frequency approximation of the discrete Sommerfeld diffraction problems, involving the scattering of a time harmonic lattice wave incident on square lattice by a discrete Dirichlet or a discrete Neumann half-plane, is investigated. It is established that the exact solution of the discrete model converges to the solution of the continuum model, i.e. the continuous Sommerfeld problem, in certain discrete Sobolev space defined by W. Hackbusch. The proof of convergence has been provided for both types of boundary conditions when the imaginary part of incident wavenumber is positive.

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“…The l p approach to the space-time finite difference schemes for deterministic parabolic and elliptic equations is well developed. The interior l 2 estimates for elliptic type finite difference schemes are established in [20], and for parabolic schemes in [1] (see also [19]). For p > 1 the interior l p estimate in the parabolic case is proved in [2] and in the elliptic case in [18,3].…”

Section: Introductionmentioning

confidence: 99%

On the $l_p$ stability estimates for stochastic and deterministic difference equations and their application to SPDEs and PDEs

Yastrzhembskiy¹

2019

Preprint

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In this paper we develop the lp-theory of space-time stochastic difference equations which can be considered as a discrete counterpart of N.V. Krylov's Lp-theory of stochastic partial differential equations (SPDEs). We also prove a Calderon-Zygmund type estimate for deterministic parabolic finite difference schemes with variable coefficients under relaxed assumptions on the coefficients, the initial data and the forcing term.

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Discrete Sobolev spaces and regularity of elliptic difference schemes

Cited by 8 publications

References 10 publications

Uniform Error Estimates for Navier--Stokes Flow with an Exact Moving Boundary Using the Immersed Interface Method

Uniform Error Estimates for Navier--Stokes Flow with an Exact Moving Boundary Using the Immersed Interface Method

Continuum limit of discrete Sommerfeld problems on square lattice

On the $l_p$ stability estimates for stochastic and deterministic difference equations and their application to SPDEs and PDEs

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