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

We consider a linear diffusion equation on $Ω:=\mathbb{R}^2\setminus\bar{Ω_\mathcal{O}}$, where $Ω_\mathcal{O}$ is a bounded domain. The time-dependent flux on the boundary $Γ:=\partialΩ_\mathcal{O}$ 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 $L^2([0,t];L^2(Γ))$-bound on the difference in flux on the boundary. Moreover, we derive for all $t>0$ an $L^2(Ω)$-bound and an $L^2([0,t];H^1(Ω))$-bound for the difference of the solutions to the two models.

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