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

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 ΔΓ on surfaces Γ defined as graphs of C^2 functions on a flat domain Ω⊂R^d−1 (d≥2), ΔΓu=f on Γ,u=0 on ∂Γ. Based on some properties of differential geometry, we transformed the Laplace Beltrami operator on the surface Γ to get an elliptic operator on the flat domain Ω, −div(A(∇u)T)=F in Ω,u=0 on ∂Ω. 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 Ω and do not involve the geometric property of the surface Γ. A numerical experiment for the AFEM algorithm is provided to illustrate the theoretical results.