Regularity Estimates up to the Boundary for Elliptic Systems of Difference Equations

Strikwerda, John C.; Wade, Bruce A.; Bube, Kenneth P.

doi:10.1137/0727020

Cited by 6 publications

(4 citation statements)

References 12 publications

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“…To analyze the general hybrid scheme, we will take the viewpoint of our previous work [22] and regard the scheme as a nonlinear finite difference scheme. One of the main ingredients we will use in this paper is the regularity estimates up to the boundary for elliptic difference systems established in [29]. We also note that the regularity estimates of elliptic difference equations and systems have been investigated by several works [6,17,18,20,24,29,32].…”

Section: Regularity Estimate Of Hybrid Schemesmentioning

confidence: 99%

“…One of the main ingredients we will use in this paper is the regularity estimates up to the boundary for elliptic difference systems established in [29]. We also note that the regularity estimates of elliptic difference equations and systems have been investigated by several works [6,17,18,20,24,29,32]. For reader's convenience, we recall here briefly the setup and results with some adaptation to the current work.…”

Section: Regularity Estimate Of Hybrid Schemesmentioning

confidence: 99%

“…To be consistent with[22], our scaling of the reciprocal space in terms of ε is slightly different from that of[29].…”

mentioning

confidence: 91%

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Stability Of A Force-Based Hybrid Method With Planar Sharp Interface

Lu¹,

Ming²

2014

SIAM J. Numer. Anal.

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Abstract. We study a force-based hybrid method that couples atomistic model with CauchyBorn elasticity model with sharp transition interface. We identify stability conditions that guarantee the convergence of the hybrid scheme to the solution of the atomistic model with second order accuracy, as the ratio between lattice parameter and the characteristic length scale of the deformation tends to zero. Convergence is established for hybrid schemes with planar sharp interface for system without defects, with general finite range atomistic potential and simple lattice structure. The key ingredient of the proof is regularity and stability analysis of elliptic systems of difference equations. We apply the results to atomistic-to-continuum scheme for a 2D triangular lattice with planar interface.Key words. Multiscale method, atomistic-to-continuum, stability analysis, force-based coupling AMS subject classifications. 65N12; 74S301. Introduction. Multiscale methods couple together atomistic and continuum models have received intense investigations in recent years; see, e.g., [1,11,23,30,31]. Generally speaking, there are two main categories of methods coupling atomistic and continuum models: energy-based methods and force-based methods. The energybased methods employ an energy that is a mixture of atomistic energy and continuum elastic energy. The energy functional is then minimized subject to suitable boundary conditions to obtain the deformed state of the system. The force-based methods work instead at the level of force balance equations. The forces derived from atomistic and continuum models are coupled together. The force balance equations supplemented with suitable boundary conditions are solved to obtain the deformed state of the system.From a numerical analysis point of view, the key issue for these multiscale methods is the consistency and stability analysis of the coupled schemes [11, Chapter 7]. In this paper, we study force-based atomistic-to-continuum hybrid methods in two and three dimension with sharp transition between the atomistic and continuum regions. In our previous work [22], we developed the stability analysis in general dimension for a force-based atomistic-to-continuum method with smooth transition between the two regions. The main focus of the current paper is to extend the stability analysis of [22] to hybrid schemes with sharp interface between atomistic and continuum models.Comprehensive reviews for force-based hybrid methods can be found in [25, Section 5 and Section 6] and [30, Section 12.5]. A class of force-based methods uses a handshake region (transition region) between the atomistic and continuum regions.

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Section: Regularity Estimate Of Hybrid Schemesmentioning

confidence: 99%

Section: Regularity Estimate Of Hybrid Schemesmentioning

confidence: 99%

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Stability Of A Force-Based Hybrid Method With Planar Sharp Interface

Lu¹,

Ming²

2014

SIAM J. Numer. Anal.

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“…In [14], Thomee also established Schauder estimates for discrete elliptic operators with smooth coefficients. In [13], regularity estimates up to the boundary for elliptic systems were obtained, again with the requirement of smooth coefficients. In [4,5], regularity results similar in spirit to those of this paper are derived in Sobolev and Hölder norms for bounded domains with Dirichlet boundary conditions, with less smoothness required of the coefficients but significantly more complicated assumptions on the discrete elliptic operator.…”

Section: Introductionmentioning

confidence: 99%

Maximum Norm Regularity of Periodic Elliptic Difference Operators With Variable Coefficients

Pruitt

2015

ESAIM: M2AN

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We prove regularity results for divergence form periodic second order elliptic difference operators on the space of functions of mean value zero, valid in maximum norm. The estimates obtained are discrete analogues of the regularity results for continuous operators. The maximum norms of the inverse of such an elliptic operator and of its first spatial differences are uniformly bounded in the grid spacing, and second spatial differences are uniformly bounded except for a logarithmic factor in the grid spacing.

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A Domain Decomposition Method for Incompressible Viscous Flow

Strikwerda¹,

Scarbnick²

1993

SIAM J. Sci. Comput.

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Regularity Estimates up to the Boundary for Elliptic Systems of Difference Equations

Cited by 6 publications

References 12 publications

Stability Of A Force-Based Hybrid Method With Planar Sharp Interface

Stability Of A Force-Based Hybrid Method With Planar Sharp Interface

Maximum Norm Regularity of Periodic Elliptic Difference Operators With Variable Coefficients

A Domain Decomposition Method for Incompressible Viscous Flow

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