In this paper we study a system of advection-diffusion equations in a bulk domain coupled to an advection-diffusion equation on an embedded surface. Such systems of coupled partial differential equations arise in, for example, the modeling of transport and diffusion of surfactants in two-phase flows. The model considered here accounts for adsorption-desorption of the surfactants at a sharp interface between two fluids and their transport and diffusion in both fluid phases and along the interface. The paper gives a well-posedness analysis for the system of bulk-surface equations and introduces a finite element method for its numerical solution. The finite element method is unfitted, i.e., the mesh is not aligned to the interface. The method is based on taking traces of a standard finite element space both on the bulk domains and the embedded surface. The numerical approach allows an implicit definition of the surface as the zero level of a level-set function. Optimal order error estimates are proved for the finite element method both in the bulk-surface energy norm and the -norm. The analysis is not restricted to linear finite elements and a piecewise planar reconstruction of the surface, but also covers the discretization with higher order elements and a higher order surface reconstruction.
DOI : 10.1051/m2an/2015013
Mots-clés : Finite element method, surface PDEs, surface-bulk coupled problems, unfitted method, transport-diffusion
@article{M2AN_2015__49_5_1303_0,
author = {Gross, Sven and Olshanskii, Maxim A. and Reusken, Arnold},
title = {A trace finite element method for a class of coupled bulk-interface transport problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1303--1330},
publisher = {EDP-Sciences},
volume = {49},
number = {5},
year = {2015},
doi = {10.1051/m2an/2015013},
zbl = {1329.76171},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015013/}
}
TY - JOUR AU - Gross, Sven AU - Olshanskii, Maxim A. AU - Reusken, Arnold TI - A trace finite element method for a class of coupled bulk-interface transport problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1303 EP - 1330 VL - 49 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015013/ DO - 10.1051/m2an/2015013 LA - en ID - M2AN_2015__49_5_1303_0 ER -
%0 Journal Article %A Gross, Sven %A Olshanskii, Maxim A. %A Reusken, Arnold %T A trace finite element method for a class of coupled bulk-interface transport problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1303-1330 %V 49 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015013/ %R 10.1051/m2an/2015013 %G en %F M2AN_2015__49_5_1303_0
Gross, Sven; Olshanskii, Maxim A.; Reusken, Arnold. A trace finite element method for a class of coupled bulk-interface transport problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1303-1330. doi : 10.1051/m2an/2015013. http://www.numdam.org/articles/10.1051/m2an/2015013/
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