Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I: The one-dimensional case

Travis Askham; Leslie Greengard

doi:10.1137/130922951

Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I: The one-dimensional case

Travis Askham, Leslie Greengard

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various L^p norms. We show that high-order accurate Nyström discretization leads to well-conditioned finite-dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen L^p space and adaptively refined in the internal layer.

Original language	English (US)
Pages (from-to)	625-641
Number of pages	17
Journal	SIAM Review
Volume	56
Issue number	4
DOIs	https://doi.org/10.1137/130922951
State	Published - 2014
Externally published	Yes

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Computational Mathematics
Applied Mathematics

Keywords

Adaptive discretization
Divergence-form elliptic equations
Integral equations
Integral operator norms
Internal layers

Access to Document

10.1137/130922951

Cite this

@article{da83337704f74d2e82367b79ac2841f2,

title = "Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I: The one-dimensional case",

abstract = "We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various Lp norms. We show that high-order accurate Nystr{\"o}m discretization leads to well-conditioned finite-dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen Lp space and adaptively refined in the internal layer.",

keywords = "Adaptive discretization, Divergence-form elliptic equations, Integral equations, Integral operator norms, Internal layers",

author = "Travis Askham and Leslie Greengard",

note = "Publisher Copyright: {\textcopyright} 2014 Society for Industrial and Applied Mathematics.",

year = "2014",

doi = "10.1137/130922951",

language = "English (US)",

volume = "56",

pages = "625--641",

journal = "SIAM Review",

issn = "0036-1445",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "4",

}

TY - JOUR

T1 - Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I

T2 - The one-dimensional case

AU - Askham, Travis

AU - Greengard, Leslie

PY - 2014

Y1 - 2014

N2 - We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various Lp norms. We show that high-order accurate Nyström discretization leads to well-conditioned finite-dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen Lp space and adaptively refined in the internal layer.

AB - We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various Lp norms. We show that high-order accurate Nyström discretization leads to well-conditioned finite-dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen Lp space and adaptively refined in the internal layer.

KW - Adaptive discretization

KW - Divergence-form elliptic equations

KW - Integral equations

KW - Integral operator norms

KW - Internal layers

UR - http://www.scopus.com/inward/record.url?scp=84908891798&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908891798&partnerID=8YFLogxK

U2 - 10.1137/130922951

DO - 10.1137/130922951

M3 - Article

AN - SCOPUS:84908891798

SN - 0036-1445

VL - 56

SP - 625

EP - 641

JO - SIAM Review

JF - SIAM Review

IS - 4

ER -

Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I: The one-dimensional case

Abstract

All Science Journal Classification (ASJC) codes

Keywords

Access to Document

Other files and links

Fingerprint

Cite this