Advanced Search

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 $x\mapsto f\left(x\right)/x$ increases on $[0,1]$. 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 ${]0,L[}^2$ with adequate non-homogeneous Dirichlet conditions. We show that the error $u_L-u$ tends to zero exponentially fast, in the uniform norm.

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