A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator

作者：Burman Erik; Han**o Peter^*; Larson Mats G

来源：Computer Methods in Applied Mechanics and Engineering, 2015, 285: 188-207.

DOI：10.1016/j.cma.2014.10.044

• Laplace-Beltrami
• Embedded surface
• Tangential calculus

We consider solving the Laplace-Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.
The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method.

• 出版日期2015-3-1