On the condition number of the finite element method for the Laplace-Beltrami operator

We give an upper bound for the condition number of the finite element operator for the Laplace-Beltrami operator on closed surfaces immersed in R-3. The expression is similar to the condition number of the Laplace operator in the Euclidean case, with the curvature affecting the condition number through the Poincare constant. However in the case of closed surfaces the finite element matrix is singular and the linear system is solved for a unique solution with zero mean. As an application, numerical simulation of the Poisson problem on a sphere is presented in this paper, and motivate the search for efficient preconditioners.