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 journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)625-641
Number of pages17
JournalSIAM Review
Volume56
Issue number4
DOIs
StatePublished - 2014

Fingerprint

Internal Layers
Elliptic PDE
Partial differential equations
Integral equations
Linear systems
Integral Equations
Discretization
Norm
Lp-norm
Lp Spaces
Elliptic Partial Differential Equations
Condition number
Variable Coefficients
One Dimension
Linear Systems
Higher Order
If and only if
Formulation

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Theoretical Computer Science

Cite this

Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I : The one-dimensional case. / Askham, Travis; Greengard, Leslie.

In: SIAM Review, Vol. 56, No. 4, 2014, p. 625-641.

Research output: Contribution to journalArticle

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