Approximation of a semilinear elliptic problem in an unbounded domain

Messaoud Kolli; Michelle Schatzman

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 1, page 117-132
  • ISSN: 0764-583X

Abstract

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Let f be an odd function of a class C 2 such that f ( 1 ) = 0 , f ' ( 0 ) < 0 , f ' ( 1 ) > 0 and x ↦ f ( x ) / x increases on [ 0 , 1 ] . We approximate the positive solution of - Δ u + f ( u ) = 0 , on ℝ + 2 with homogeneous Dirichlet boundary conditions by the solution of - Δ u L + f ( u L ) = 0 , on ] 0 , L [ 2 with adequate non-homogeneous Dirichlet conditions. We show that the error u L - u tends to zero exponentially fast, in the uniform norm.

How to cite

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Kolli, Messaoud, and Schatzman, Michelle. "Approximation of a semilinear elliptic problem in an unbounded domain." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.1 (2003): 117-132. <http://eudml.org/doc/245741>.

@article{Kolli2003,
abstract = {Let $f$ be an odd function of a class $\mathrm \{C\}^\{2\}$ such that $f(1)=0,f^\{\prime \}(0)&lt;0,f^\{\prime \}(1)&gt;0$ and $x\mapsto f(x)/x$ increases on $[0,1]$. We approximate the positive solution of $-\Delta u+f(u)=0,$ on $\mathbb \{R\}_\{+\}^\{2\}$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_\{L\}+f(u_\{L\})=0,$ on $]0,L[^\{2\}$ with adequate non-homogeneous Dirichlet conditions. We show that the error $u_\{L\}-u$ tends to zero exponentially fast, in the uniform norm.},
author = {Kolli, Messaoud, Schatzman, Michelle},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {semilinear elliptic equations; full-space problems; approximation by finite domains},
language = {eng},
number = {1},
pages = {117-132},
publisher = {EDP-Sciences},
title = {Approximation of a semilinear elliptic problem in an unbounded domain},
url = {http://eudml.org/doc/245741},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Kolli, Messaoud
AU - Schatzman, Michelle
TI - Approximation of a semilinear elliptic problem in an unbounded domain
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 1
SP - 117
EP - 132
AB - Let $f$ be an odd function of a class $\mathrm {C}^{2}$ such that $f(1)=0,f^{\prime }(0)&lt;0,f^{\prime }(1)&gt;0$ and $x\mapsto f(x)/x$ increases on $[0,1]$. We approximate the positive solution of $-\Delta u+f(u)=0,$ on $\mathbb {R}_{+}^{2}$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_{L}+f(u_{L})=0,$ on $]0,L[^{2}$ with adequate non-homogeneous Dirichlet conditions. We show that the error $u_{L}-u$ tends to zero exponentially fast, in the uniform norm.
LA - eng
KW - semilinear elliptic equations; full-space problems; approximation by finite domains
UR - http://eudml.org/doc/245741
ER -

References

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