Approximation of a semilinear elliptic problem in an unbounded domain
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003) no. 1 , p. 117-132
doi : 10.1051/m2an:2003017
URL stable : http://www.numdam.org/item?id=M2AN_2003__37_1_117_0

Classification:  35J60,  35P15
Let $f$ be an odd function of a class ${\mathrm{C}}^{2}$ such that $f\left(1\right)=0,{f}^{\text{'}}\left(0\right)<0,{f}^{\text{'}}\left(1\right)>0$ and $x↦f\left(x\right)/x$ increases on $\left[0,1\right]$. We approximate the positive solution of $-\Delta u+f\left(u\right)=0,$ on ${ℝ}_{+}^{2}$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta {u}_{L}+f\left({u}_{L}\right)=0,$ on ${\right]0,L\left[}^{2}$ with adequate non-homogeneous Dirichlet conditions. We show that the error ${u}_{L}-u$ tends to zero exponentially fast, in the uniform norm.

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