Dimension Reduction and Error Estimates

Abstract

The aim of this article is twofold: to provide asymptotic behavior and error estimates results for an elliptic dimension reduction problem. The problem is posed in a bounded and thin domain ${\mathrm{\Omega }}_{\delta }\doteq \mathcal{O}\times {\omega }_{\delta }\subset {\mathbb{R}}^m\times {\mathbb{R}}^n$ , $m,n\ge 1$ where ${\omega }_{\delta }=\delta \omega$ when $\delta \to 0$ (the thickness of ${\mathrm{\Omega }}_{\delta }$ is of order $\delta$ while its diameter is of order 1). The limit problem is elliptic and is posed in $\mathcal{O}$ . Under the assumptions that $\mathcal{O}$ has a smooth boundary $(C^{1,1})$ and homogeneous Dirichlet boundary condition, we give two important results, first an estimate of the global error (of order ${\delta }^{1/2}$ ), and then an estimate of the $L^2$ global and $H^1$ interior errors (of order $\delta$ ).