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Efficient Spectral Methods for PDEs with Spectral Fractional Laplacian

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Abstract

We develop efficient spectral methods for the spectral fractional Laplacian equation and parabolic PDEs with spectral fractional Laplacian on rectangular domains. The key idea is to construct eigenfunctions of discrete Laplacian (also referred to Fourier-like basis) by using the Fourierization method. Under this basis, the non-local fractional Laplacian operator can be trivially evaluated, leading to very efficient algorithms for PDEs involving spectral fractional Laplacian. We provide a rigorous error analysis of the proposed methods for the case with homogeneous boundary conditions, as well as ample numerical results to show their effectiveness.

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Authors and Affiliations

  1. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Changtao Sheng

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-1957, USA

    Duo Cao & Jie Shen

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  1. Changtao Sheng
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  2. Duo Cao
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  3. Jie Shen
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Correspondence to Jie Shen.

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The work of J. Shen is partially supported by NSF DMS-2012585 and AFOSR FA9550-20-1-0309.

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Sheng, C., Cao, D. & Shen, J. Efficient Spectral Methods for PDEs with Spectral Fractional Laplacian. J Sci Comput 88, 4 (2021). https://doi.org/10.1007/s10915-021-01491-2

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  • DOI: https://doi.org/10.1007/s10915-021-01491-2

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