Approximation of the spectral fractional powers of the Laplace-Beltrami Operator

13 Jan 2021  ·  Andrea Bonito, Wenyu Lei ·

We consider numerical approximations of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consist of a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rates of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in $L^2$ and $H^1$. The performances of the algorithms are illustrated in different settings including the approximation of Gaussian fields on surfaces.

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Numerical Analysis Numerical Analysis 65M12, 65M15, 65M60, 35S11, 65R20