A trace finite element method for a class of coupled bulk-interface transport problems
Gross, Sven1 ; Olshanskii, Maxim A.2 ; Reusken, Arnold1
1 Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, 52056 Aachen, Germany
2 Department of Mathematics, University of Houston, Houston, TX-77204-3008, USA
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1303-1330.

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 L2-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.

Reçu le :
DOI : 10.1051/m2an/2015013
Classification : 65N30, 65N15, 76T99
Mots-clés : Finite element method, surface PDEs, surface-bulk coupled problems, unfitted method, transport-diffusion
Gross, Sven 1 ; Olshanskii, Maxim A. 2 ; Reusken, Arnold 1

1 Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, 52056 Aachen, Germany
2 Department of Mathematics, University of Houston, Houston, TX-77204-3008, USA 
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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. https://www.numdam.org/articles/10.1051/m2an/2015013/

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