A conservative Eulerian finite element method for transport and diffusion in moving domains
Abstract
The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from Lehrenfeld & Olshanskii [ESAIM: M2AN, 53(2): 585-614, 2019] of a solution extension to realise the Eulerian time-stepping scheme. However, a reformulation of the partial differential equation is suggested to derive a scheme which conserves the quantity under consideration exactly on the discrete level. For the spatial discretisation, the paper considers an unfitted finite element method. Ghost-penalty stabilisation is used to realise the discrete solution extension and gives a scheme robust against arbitrary intersections between the mesh and geometry interface. The stability is analysed for both first- and second-order backward differentiation formula versions of the scheme. Several numerical examples in two and three spatial dimensions are included to illustrate the potential of this method.
Keywords
Cite
@article{arxiv.2404.07130,
title = {A conservative Eulerian finite element method for transport and diffusion in moving domains},
author = {Maxim Olshanskii and Henry von Wahl},
journal= {arXiv preprint arXiv:2404.07130},
year = {2025}
}