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This paper focuses on numerical approximation for fractional powers ofelliptic operators on $2$-d manifolds. Firstly, parametric finite elementmethod is employed to discretize the original problem. We then approximatefractional powers of the discrete elliptic operator by the product of rationalfunctions, each of which is a diagonal Pad\'e approximant for correspondingpower function. Rigorous error analysis is carried out and sharp error boundsare presented which show that the scheme is robust for $\alpha\rightarrow 0^+$and $\alpha \rightarrow 1^-$. The cost of the proposed algorithm is solvingsome elliptic problems. Since the approach is exponentially convergent withrespect to the number of solves, it is very efficient. Some numerical tests aregiven to confirm our theoretical analysis and the robustness of the algorithm.

Pad\'e-parametric FEM approximation for fractional powers of elliptic operators on manifolds