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A posteriori error estimations of the Petrov-Galerkin methods for fractional Helmholtz equations

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Abstract

Here, we develop a Petrov-Galerkin spectral method for the fractional Helmholtz equations (FHEs) of even order ν = s + k,s ∈ (k − 1,k) and k ∈ℕ. We define trial and test functions by related generalized Jacobi functions (GJFs). Moreover, we efficiently establish the well-posedness of problem and provide the rigorous priori error estimates. Furthermore, by auxiliary equation, we also obtain the super-approximation estimates. Notably, we propose a post-processing technique for the Petrov-Galerkin spectral method, and give the error estimates of corrected solution. In addition, we define a posteriori error estimators, and prove that they are asymptotically accurate. Finally, we demonstrate the sharpness of our error estimates by numerical experiments.

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Funding

This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (41974133, 11671157).

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, Shandong, People’s Republic of China

    Wenting Mao

  2. School of Mathematical Science, South China Normal University, Guangzhou, 520631, Guangdong, People’s Republic of China

    Yanping Chen

  3. School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, People’s Republic of China

    Huasheng Wang

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  1. Wenting Mao
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  3. Huasheng Wang
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Correspondence to Yanping Chen.

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Mao, W., Chen, Y. & Wang, H. A posteriori error estimations of the Petrov-Galerkin methods for fractional Helmholtz equations. Numer Algor 89, 1095–1127 (2022). https://doi.org/10.1007/s11075-021-01147-0

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