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Volume 15, Issue 4
Approximation of the Spectral Fractional Powers of the Laplace-Beltrami Operator

Andrea Bonito & Wenyu Lei

DOI: 10.4208/nmtma.OA-2022-0005s

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 1193-1218.

Published online: 2022-10

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  • Abstract

We consider numerical approximation of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consists a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rate 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 on different settings including the approximation of Gaussian fields on surfaces.

  • Keywords

Fractional diffusion, Laplace-Beltrami, FEM parametric methods on surfaces, Gaussian fields.

  • AMS Subject Headings

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

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-1193, author = {Bonito , Andrea and Lei , Wenyu}, title = {Approximation of the Spectral Fractional Powers of the Laplace-Beltrami Operator}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {1193--1218}, abstract = {

We consider numerical approximation of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consists a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rate 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 on different settings including the approximation of Gaussian fields on surfaces.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0005s}, url = {http://global-sci.org/intro/article_detail/nmtma/21099.html} }
TY - JOUR T1 - Approximation of the Spectral Fractional Powers of the Laplace-Beltrami Operator AU - Bonito , Andrea AU - Lei , Wenyu JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1193 EP - 1218 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0005s UR - https://global-sci.org/intro/article_detail/nmtma/21099.html KW - Fractional diffusion, Laplace-Beltrami, FEM parametric methods on surfaces, Gaussian fields. AB -

We consider numerical approximation of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consists a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rate 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 on different settings including the approximation of Gaussian fields on surfaces.

Andrea Bonito & Wenyu Lei. (2022). Approximation of the Spectral Fractional Powers of the Laplace-Beltrami Operator. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 1193-1218. doi:10.4208/nmtma.OA-2022-0005s
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