Application of Adaptive Finite Element Method for Elliptic Partial Differential Equations to the Laplace Beltrami Operator on Graphs

作者：Mekchay Khamron^*

来源：Bulletin of the Malaysian Mathematical Sciences Society, 2012, 35(2A): 509-518.

• Laplace-Beltrami operator
• graph
• adaptive finite element method
• elliptic PDE

The Laplace Beltrami operator, known as an elliptic operator for functions defined on surfaces, appears in some applications in sciences and engineerings. In this paper we consider the Laplace Beltrami operator Delta(Gamma) on surfaces Gamma defined as graphs of C-2 functions on a flat domain Omega subset of Rd-1 (d %26gt;= 2), %26lt;br%26gt;Delta(Gamma)u = f on Gamma, u = 0 on partial derivative Gamma. %26lt;br%26gt;Based on some properties of differential geometry, we transformed the Laplace Beltrami operator on the surface Gamma to get an elliptic operator on the flat domain Omega, %26lt;br%26gt;-div(A(del u)(T)) = F in Omega. u = 0 on partial derivative Omega. %26lt;br%26gt;We applied an adaptive finite element method (AFEM) for a general second order linear elliptic partial differential equations developed by K. Mekchay and R. H. Nochetto to solve the transformed problem. The a posteriori error estimates in energy norm and the design or algorithm are derived accordingly for the transformed problem in the sense of the elliptic operator. The discretization and mesh generations of the AFEM algorithm rely on indicators and oscillations which now depend on the data A and F of the elliptic operator on Omega and do not involve the geometric property of the surface Gamma. A numerical experiment for the AFEM algorithm is provided to illustrate the theoretical results.

• 出版日期2012