|Preprint-No.:||< 398 >||Published in:||July 2014||PDF-File:||IGPM398.pdf|
|Title:||A Trace Finite Element Method for a Class of Coupled Bulk-Interface Transport Problemes|
|Authors:||Sven Groß, Maxim A. Olshanskii, Arnold Reusken|
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.
|Keywords:||coupled bulk-surface problems, trace ﬁnite elements, surfactant transport|
|Publication:||ESAIM: M2AN |
Vol. 49 5, Sep.-Oct. 2015, Pages 1303 - 1330