A tensor product generalized ADI method for elliptic problems on cylindrical domains with holes

The paper is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second-order elliptic problems by Chipot and Rougirel in [On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, Commun. Contemp. Math. 4(1) (2002) 15–44], where the force functions are considered on the cross-section of domains, we prove the non-local counterpart of their result.Recently in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89(1–2) (2014) 21–35] Yeressian established a weighted estimate for solutions of non-local Dirichlet problems which exhibit the asymptotic behavior. The case when [Formula: see text] was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this paper, we extend this result to each order between [Formula: see text] and [Formula: see text].

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Hierarchical Tensor-Product Approximation to the Inverse and Related Operators for High-Dimensional Elliptic Problems

Computing ◽

10.1007/s00607-004-0086-y ◽

2004 ◽

Vol 74 (2) ◽

pp. 131-157 ◽

Cited By ~ 53

Author(s):

Ivan P. Gavrilyuk ◽

Wolfgang Hackbusch ◽

Boris N. Khoromskij

Keyword(s):

Tensor Product ◽

Elliptic Problems ◽

High Dimensional ◽

Tensor Product Approximation

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The Dynamical Approach to Elliptic Problems in Cylindrical Domains, and a Study of Their Parabolic Singular Limit

Journal of Differential Equations ◽

10.1006/jdeq.1993.1030 ◽

1993 ◽

Vol 102 (2) ◽

pp. 244-304 ◽

Cited By ~ 31

Author(s):

A. Calsina ◽

X. Mora ◽

J. Solamorales

Keyword(s):

Elliptic Problems ◽

Singular Limit ◽

Dynamical Approach ◽

Cylindrical Domains

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ON THE ASYMPTOTIC BEHAVIOUR OF THE SOLUTION OF ELLIPTIC PROBLEMS IN CYLINDRICAL DOMAINS BECOMING UNBOUNDED

Communications in Contemporary Mathematics ◽

10.1142/s0219199702000555 ◽

2002 ◽

Vol 04 (01) ◽

pp. 15-44 ◽

Cited By ~ 17

Author(s):

M. CHIPOT ◽

A. ROUGIREL

Keyword(s):

Asymptotic Behavior ◽

Asymptotic Behaviour ◽

Cross Section ◽

Elliptic Problems ◽

Nonlinear Elliptic Problems ◽

Nonlinear Elliptic ◽

Cylindrical Domains ◽

Linear And Nonlinear ◽

The Cross

We study the asymptotic behavior of the solution of linear and nonlinear elliptic problems in cylindrical domains becoming unbounded in one or several directions. In particular we show that this solution converges in H1 norm toward the solution of problems set on the cross section of the domains.

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