2014/06/30 by Sven Groß, Gross, Sven, Maxim A. Olshanskii +3 · 2 citations
Computer Science · Engineering · Mathematics · #65N30 #76T10 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1406.7694
openalex publication_date 2014/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.