Modelling with measures: Approximation of a mass-emitting object by a point source

Abstract

We consider a linear diffusion equation on $\Omega :={\mathbb{R}}^2\setminus \overline{{\Omega }_{\mathcal{O}}}$, where ${\Omega }_{\mathcal{O}}$ is a bounded domain. The time-dependent flux on the boundary $\Gamma :=\partial {\Omega }_{\mathcal{O}}$ is prescribed. The aim of the paper is to approximate the dynamics by the solution of the diffusion equation on the whole of ${\mathbb{R}}^2$ with a measure-valued point source in the origin and provide estimates for the quality of approximation. For all time $t$, we derive an $L^2([0,t];L^2(\Gamma ))$-bound on the difference in flux on the boundary. Moreover, we derive for all $t>0$ an $L^2(\Omega )$-bound and an $L^2([0,t];H^1(\Omega ))$-bound for the difference of the solutions to the two models.