Stabilized CutFEM for the Convection Problem on Surfaces
This work provides a stable and accurate numerical method for solving convection problems on surfaces, which is important for applications in computational geometry and fluid dynamics.
The paper develops a stabilized cut finite element method for convection problems on surfaces, achieving h^{3/2} convergence in the natural norm and h^{3/4} convergence for the full gradient in L^2, with condition number bounded by h^{-2}.
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.