Asymptotic expansions of lattice Green's functions

Martinsson, Per-Gunnar; Rodin, Gregory J.

doi:10.1098/rspa.2002.0985

Cited by 50 publications

(55 citation statements)

References 13 publications

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“…The methods for the free space problem, and for lattices with inclusions, apply directly once a fundamental solution for the operator under consideration has been constructed. Techniques for constructing such fundamental solutions are described in [12]. The techniques for handling boundary conditions in Section 4 also generalize, with the only caveat that for difference operators that involve more than the eight nearest neighbors of any lattice node, the boundary of a lattice domain must be extended to a boundary layer of nodes, sufficiently wide that the nodes inside the layer do not communicate directly with the nodes on the outside.…”

Section: Generalization To Other Lattice Operatorsmentioning

confidence: 98%

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Fast and accurate numerical methods for solving elliptic difference equations defined on lattices

Gillman

Martinsson

2010

Journal of Computational Physics

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Discrete Laplace operator Boundary integral equation Fast solver Fast direct solver Fast multipole method Hierarchically semi-separable matrix a b s t r a c t Techniques for rapidly computing approximate solutions to elliptic PDEs such as Laplace's equation are well established. For problems involving general domains, and operators with constant coefficients, a highly efficient approach is to rewrite the boundary value problem as a Boundary Integral Equation (BIE), and then solve the BIE using fast methods such as, e.g., the Fast Multipole Method (FMM). The current paper demonstrates that this procedure can be extended to elliptic difference equations defined on infinite lattices, or on finite lattice with boundary conditions of either Dirichlet or Neumann type. As a representative model problem, a lattice equivalent of Laplace's equation on a square lattice in two dimensions is considered: discrete analogs of BIEs are derived and fast solvers analogous to the FMM are constructed. Fast techniques are also constructed for problems involving lattices with inclusions and local deviations from perfect periodicity. The complexity of the methods described is O(N boundary + N source + N inc ) where N boundary is the number of nodes on the boundary of the domain, N source is the number of nodes subjected to body loads, and N inc is the number of nodes that deviate from perfect periodicity. This estimate should be compared to the O(N domain logN domain ) estimate for FFT based methods, where N domain is the total number of nodes in the lattice (so that in two dimensions, N boundary $ N 1=2 domain ). Several numerical examples are presented.

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Section: Generalization To Other Lattice Operatorsmentioning

confidence: 98%

“…[8,[12][13][14][15]) that as jmj ? 1, the fundamental solution / defined by (2.13) has the asymptotic expansion…”

Section: Asymptotic Expansionmentioning

confidence: 99%

Fast and accurate numerical methods for solving elliptic difference equations defined on lattices

Gillman

Martinsson

2010

Journal of Computational Physics

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“…Our principal motivation in this paper is to focus on the adaptive algorithm rather than modeling issues. Various techniques have been suggested to derive continuum models, see for example [3,7]. For the lattice model (8), in the limit as H → 0, we consider here the following continuum model of the problem:…”

Section: A Continuum Surrogate Problemmentioning

confidence: 99%

On the Extension of Goal-Oriented Error Estimation and Hierarchical Modeling to Discrete Lattice Models

Oden¹,

Prudhomme²,

Bauman³

2004

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The Goals algorithm for adaptive modeling is extended to the case of discrete models such as those characterized by lattice structures (or the static behavior molecular systems) by using surrogate models obtained from continuum approximations of the lattice. The result is a technique that could provide for scale-bridging between continuum models and atomistic models, although the present development concerns only simple algebraic systems. An example is provided in which quantities of interest in a large number of degrees of freedom system are computed to a preset tolerance in relatively few low-order approximations.

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“…The method is to regard the solution of the continuous problem as an approximate solution to the discrete problem. An alternative approach based on comparison of the Fourier representation (4.17) of the lattice Green's function G lattice a hom (·, ·) to the Fourier representation of the continuous Green's function G a hom (·, ·) is pursued in [22] for the case of elliptic equations.…”

Section: Fluctuations Of Averaged Green's Functionsmentioning

confidence: 99%

Strong convergence to the homogenized limit of parabolic equations with random coefficients

Conlon

Fahim

2014

Trans. Amer. Math. Soc.

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Abstract. This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing.

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Asymptotic expansions of lattice Green's functions

Abstract: We derive asymptotic expansions for the Green functions associated with coercive difference equations on general lattices. These expansions lead to rigorous methods for approximating the lattice Green functions by polyharmonic rational functions.

Cited by 50 publications

References 13 publications

Fast and accurate numerical methods for solving elliptic difference equations defined on lattices

Fast and accurate numerical methods for solving elliptic difference equations defined on lattices

On the Extension of Goal-Oriented Error Estimation and Hierarchical Modeling to Discrete Lattice Models

Strong convergence to the homogenized limit of parabolic equations with random coefficients

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