953 research outputs found
A Trace Finite Element Method for Vector-Laplacians on Surfaces
We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D
domain, which results from the modeling of surface fluids based on exterior
Cartesian differential operators. The main topic of this paper is the
development and analysis of a finite element method for the discretization of
this surface partial differential equation. We apply the trace finite element
technique, in which finite element spaces on a background shape-regular
tetrahedral mesh that is surface-independent are used for discretization. In
order to satisfy the constraint that the solution vector field is tangential to
the surface we introduce a Lagrange multiplier. We show well-posedness of the
resulting saddle point formulation. A discrete variant of this formulation is
introduced which contains suitable stabilization terms and is based on trace
finite element spaces. For this method we derive optimal discretization error
bounds. Furthermore algebraic properties of the resulting discrete saddle point
problem are studied. In particular an optimal Schur complement preconditioner
is proposed. Results of a numerical experiment are included
A trace finite element method for a class of coupled bulk-interface transport problems
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
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