Volume 49, Number 5, September-October 2015
|Page(s)||1303 - 1330|
|Published online||18 August 2015|
A trace finite element method for a class of coupled bulk-interface transport problems∗
1 Institut für Geometrie und Praktische
Mathematik, RWTH-Aachen University, 52056
2 Department of Mathematics, University of Houston, Houston, TX - 77204-3008, USA
Revised: 12 February 2015
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.
Mathematics Subject Classification: 65N30 / 65N15 / 76T99
Key words: Finite element method / surface PDEs / surface-bulk coupled problems / unfitted method / transport-diffusion
© EDP Sciences, SMAI 2015
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