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

Joep H.M. Evers; Sander C. Hille; Adrian Muntean

doi:10.3934/mbe.2015.12.357

Article Contents

2015, Volume 12, Issue 2: 357-373. Doi: 10.3934/mbe.2015.12.357 open access

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

1.
Institute for Complex Molecular Systems & Centre for Analysis, Scientific computing and Applications, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven
2.
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden

Received: March 31, 2014

Revised: September 30, 2014

Published: November 2014

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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 $\Gamma:=∂\Omega_\mathcal{o}$ 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.

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Mathematics Subject Classification: Primary: 35K05, 35A35; Secondary: 35B45.

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