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A Posteriori Error Estimates of Spectral Approximations for Second Order Partial Differential Equations in Spherical Geometries

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Abstract

In this paper, we investigate a posteriori error estimates of the Galerkin spectral methods for second-order equations, and propose a simple type of error estimator comprising expansion coefficients of known quantities such as the right-hand term. We first show that the errors of the numerical solution of the Poisson equation on the unit ball in arbitrary dimensions can be identified by the approximation errors of the (weighted) \(L^2\)-projection of the right-hand function together with the non-homogeneous boundary function. This result indicates that the decay rate of the high frequency coefficients of the right-hand term in weighted orthogonal ball polynomials and of the boundary term in spherical harmonics serves as an ideal a posteriori error estimator. In the sequel, we establish a posteriori error estimates on the Galerkin spectral method applied to the singular perturbation problem of a reaction–diffusion equation on the unit ball. Again, the efficiency is given by the approximation errors of the weighted \(L^2\)-projection of the right-hand function; while the reliability is determined by the truncation errors of the right-hand function together with exponentially decaying multiples of the low frequency coefficients, which also reveals that the a posterior error estimator is dominated by the decay rate of the high frequency coefficients of the right-hand term. Finally, numerical examples are presented to illustrate the theoretical results.

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100193, China

    Jianwei Zhou, Huiyuan Li & Zhimin Zhang

  2. School of Mathematics and Statistics, Linyi University, Shandong, 276005, China

    Jianwei Zhou

  3. Institute of Software, Chinese Academy of Science, Beijing, 100190, China

    Huiyuan Li

  4. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

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  1. Jianwei Zhou
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  2. Huiyuan Li
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Correspondence to Huiyuan Li.

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The first author was supported in part by the NSFC (No. 11926355), NSF of Shandong (No. ZR2019YQ05, 2019KJI003), and China Postdoctoral Science Foundation (No. 2017M610751 and No. 2017T100030) The second author was supported in part by the National Key R&D Program of China (No. 2018YFB0204404) and NSFC (No. 11871145 and No. 11971016). The third author was supported in part by the NSFC (No. 11871092, No. 12131005 and NSAF U1930402).

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Zhou, J., Li, H. & Zhang, Z. A Posteriori Error Estimates of Spectral Approximations for Second Order Partial Differential Equations in Spherical Geometries. J Sci Comput 90, 56 (2022). https://doi.org/10.1007/s10915-021-01696-5

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  • DOI: https://doi.org/10.1007/s10915-021-01696-5

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