A flexible model for non-stationary Gaussian random fields on hypersurfaces is introduced.The class of random fields on curves and surfaces is characterized by an amplitude spectral density of a second order elliptic differential operator.Sampling is done by a Galerkin--Chebyshev approximation based on the surface finite element method and Chebyshev polynomials. Strong error bounds are shown with convergence rates depending on the smoothness of the approximated random field. Numerical experiments that confirm the convergence rates are presented.

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