Error estimates related to singular perturbations of first-order equations and systems
Let \(\Omega\) be a bounded open subset in \(R_ 2\), with boundary \(\Gamma\). The homogeneous Dirichlet problem \[ -\epsilon \Delta u_{\epsilon}+a\nabla u_{\epsilon}+bu_{\epsilon}=f\quad in\quad \Omega,\quad u_{\epsilon}=0\quad on\quad \Gamma, \] is studied for \(\epsilon >0\). The solution \(u_{\epsilon}\) is considered as a perturbation of the solution \(u_ 0\) of the mixed problem \(a\nabla u_ 0+bu_ 0=f\) in \(\Omega\), \(u_ 0=0\) on \(\Gamma_-\), where \(\Gamma_- ,\Gamma_+\) and \(\Gamma_ 0\) are the parts of the boundary where \(a\gamma <0\), \(a\gamma >0\) and \(a\gamma =0\), respectively (\(\gamma\) being the outward unit vector field). The authors propose an estimate \[ | u_{\epsilon}(x)-u_ 0(x)| \leq c\{\epsilon +\exp [-\alpha_+(d_+(x)/\epsilon)]+\exp (-\alpha_ 0(d_ 0(x)/\sqrt{\epsilon})]\}, \] where \(d_+\) denotes the distance from x to \(\Gamma_+\), \(d_ 0\) denotes the distance from x to \(\Gamma_ 0\cup \Sigma\) (\(\Sigma\) being the union of all characteristic curves emerging from \(\Gamma_ 0)\). The meaning of this estimate is that \(u_{\epsilon}\) converges toward \(u_ 0\) quickly, far from \(\Gamma_+\cup \Gamma_ 0\cup \Sigma\), but it also converges faster near \(\Gamma_+\) than near \(\Gamma_ 0\cup \Sigma.\) The authors are not able to prove this estimate under general assumptions. They specify, however, rather plausible conditions under which the estimate is valid. The proof is given and discussed in the paper.
scientific article; zbMATH DE number 4023806
scientific article