Abstract
We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove h^{3/2} order convergence in the natural norm associated with the method and that the full gradient enjoys h^{3/4} order of convergence in L^2. We also show that the condition number of the stiffness matrix is bounded by h^{-2}. Finally, our results are verified by numerical examples.
Highlights
In this contribution we develop a stabilized cut finite element for stationary convection on a surface embedded in R3
The method is based on a three dimensional background mesh consisting of tetrahedra and a piecewise linear approximation of the surface
We show that for the convection problem the properties of cut finite element method completely reflects the properties of the corresponding method on standard triangles or tetrahedra, see the analysis for the latter in [3]
Summary
In this contribution we develop a stabilized cut finite element for stationary convection on a surface embedded in R3. The finite element space is the continuous piecewise linear functions on the background mesh and the bilinear form defining the method only involves integrals on the surface. We note that similar stabilization terms have recently been used for stabilization of cut finite element methods for time dependent problems in [20], bulk domain problems involving standard boundary and interface conditions [4,5,18,21], and for coupled bulk–surface problems involving the Laplace–Beltrami operator on the surface in [8]. We mention [7] where a discontinuous cut finite element method for the Laplace–Beltrami operator was developed None of these references consider the convection problem on the surface. The outline of the remainder of this paper is as follows: In Sect. 2 we formulate the model problem; in Sect. 3 we define the discrete surface, its approximation properties, and the finite element method; in Sect. 4 we summarize some preliminary results involving lifting of functions from the discrete surface to the continuous surface; in Sect. 5 we first derive some technical lemmas essentially quantifying the stability induced by the stabilization term, and we derive the key discrete stability estimate; in Sect. 6 we prove a priori estimates; in Sect. 7 we prove an estimate of the condition number; and in Sect. 8, we present some numerical examples illustrating the theoretical results
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