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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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References

  1. Ainsworth, M., Oden, J.: A procedure for a posteriori error estimation for \(h\)-\(p\) finite element methods. Comput. Methods Appl. Mech. Eng. 101(1), 73–96 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ainsworth, M., Oden, J.: A posteriori error estimators for second order elliptic systems. Part 1: theoretical foundations and a posteriori error analysis. Comput. Math. Appl. 25(2), 101–113 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ainsworth, M., Oden, J.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 142(1), 1–88 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ainsworth, M., Oden, J.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2011)

    MATH  Google Scholar 

  5. Babuška, I., Rheinboldt, W.: A-posteriori error estimates for the finite element method. Int. J. Numer. Methods Eng. 12(10), 1597–1615 (1978)

    Article  MATH  Google Scholar 

  6. Bank, R., Weiser, A.: Some a posteriori error estimates for elliptic partial differential equations. Math. Comput. 44(17), 283–301 (1985)

    Article  MATH  Google Scholar 

  7. Bernardi, C., Maday, Y.: Spectral methods. In: Techniques of Scientific Computing (Part 2). Handbook of Numerical Analysis, vol. 5, pp. 209–485. Elsevier, Amsterdam (1997)

  8. Boyd, J.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover Publications, New York (2000)

    Google Scholar 

  9. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2010)

    Google Scholar 

  10. Bürg, M.: A residual-based a posteriori error estimator for the \(hp\)-finite element method for Maxwell’s equations. Appl. Numer. Math. 62(8), 922–940 (2012)

  11. Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  12. Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  13. Carstensen, C., Bartels, S., Jansche, S.: A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92(2), 233–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, M., Xu, L.: An a posteriori error estimate for a spectral method for a second-order elliptic differential equation with variable coefficients. Numer. Math. J. Chin. Univ. (English Series) 1, 60–68 (2005)

    MathSciNet  Google Scholar 

  15. Chen, Y., Huang, Y., Yi, N.: A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations. Sci. China Ser. A Math. 51, 1376–1390 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics, Springer, New York (2013)

    Book  MATH  Google Scholar 

  17. Dari, E., Duran, R., Padra, C.: Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comput. 64, 1017–1033 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ghanem, R., Sissaoui, H.: A posteriori error estimate by a spectral method of an elliptic optimal control problem. J. Comput. Math. Optim. 2, 111–125 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Gottlieb, D., Orszag, S.: Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics (1977)

  20. Guo, B.-Q.: Recent progress on a posteriori error analysis for the p and h-p finite element methods. Contemp. Math. 383, 47–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, B.-Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  22. Han, J., Zhang, Z., Yang, Y.: A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem. Numer. Methods Part. Differ. Equ. 31, 31–53 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kalnay de Rivas, E.: On the use of nonuniform grids in finite-difference equations. J. Comput. Phys. 10, 202–210 (1972)

    Article  MATH  Google Scholar 

  24. Karniadakis, G., Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2005)

    Book  MATH  Google Scholar 

  25. Li, H., Xu, Y.: Spectral approximation on the unit ball. SIAM J. Numer. Anal. 52(6), 2647–2675 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lin, R., Stynes, M.: A balanced finite element method for singularly perturbed reaction–diffusion problems. SIAM J. Numer. Anal. 50(5), 2729–2743 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, W., Ma, H., Tang, T., Yan, N.: A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations. SIAM J. Numer. Anal. 42, 1032–1061 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Melenk, J.: \(hp\)-interpolation of nonsmooth functions and an application to \(hp\)—a posteriori error estimation. SIAM J. Numer. Anal. 43(1), 127–155 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Melenk, J., Wohlmuth, B.: On residual-based a posteriori error estimation in \(hp\)-FEM. Adv. Comput. Math. 15, 311–331 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Oberkampf, W., Roy, C.: Verification and Validation in Scientific Computing. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  31. Orszag, S.A., Israeli, M.: Numerical simulation of viscous incompressible flows. Annu. Rev. Fluid Mech. 6, 281–318 (1974)

    Article  MATH  Google Scholar 

  32. Owens, R.: A posteriori error estimates for spectral element solutions to viscoelastic flow problems. Comput. Methods Appl. Mech. Eng. 164(3), 375–395 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pérez, T., Piñar, M., Xu, Y.: Weighted Sobolev orthogonal polynomials on the unit ball. J. Approx. Theory 171, 84–104 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers, with an Introduction by George E. Andrews. Narosa Publishing House, New Delhi (1988)

    MATH  Google Scholar 

  35. Sauter, S., Zech, J.: A posteriori error estimation of \(hp\)-DG finite element methods for highly indefinite Helmholtz problems. SIAM J. Numer. Anal. 53, 2414–2440 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Schwab, C.: \(p\)-and \(hp\)-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanis. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  37. Schwab, C., Suri, M.: The \(p\) and \(hp\) versions of the finite element method for problems with boundary layers. Math. Comput. 65(216), 1403–1429 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. Szegö, G.: Orthogonal Polynomials. AMS, Providence (1975)

    MATH  Google Scholar 

  39. Temme, N.: Orthogonal Polynomials. AMS Colloquium Publications, New York (1939)

    Google Scholar 

  40. Triebel, H.: Interpolation Theory Function Spaces Differential Operators. North-Holland Pub. Co., Amsterdam (1978)

    MATH  Google Scholar 

  41. Verführth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in Numerical Mathematics, Wiley-Teubner, New York (1996)

    Google Scholar 

  42. Wang, W., Xu, C.: A posteriori error estimation of spectral and spectral element methods for the Stokes/Darcy coupled problem. J. Math. Study 47, 85–110 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wiberg, N.-E., Li, X.: Super convergent patch recovery of finite-element solution and a posteriori \(l_2\) norm error estimate. Commun. Numer. Methods Eng. 10, 313–320 (1994)

    Article  MATH  Google Scholar 

  44. Yi, L., Guo, B.-Q.: A-posteriori error estimation for the Legendre spectral Galerkin method in one-dimension. Numer. Math. Theory Methods Appl. 3, 40–52 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang, J., Li, H., Wang, L.-L., Zhang, Z.: Ball prolate spheroidal wave functions in arbitrary dimensions. Appl. Comput. Harmon. Anal (2018)

  46. Zhang, Z., Zhu, J.: Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (I). Comput. Methods Appl. Mech. Eng. 123, 173–187 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhou, J., Yang, D.: An improved a posteriori error estimate for the Galerkin spectral method in one dimension. Comput. Math. Appl. 61(2), 334–340 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhou, J., Zhang, J., Jiang, Z.: The a posteriori error estimates of Chebyshev–Petrov–Galerkin methods for second-order equations. Appl. Math. Lett. 60, 126–134 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zienkiewicz, O.: The background of error estimation and adaptivity in finite element computations. Comput. Methods Appl. Mech. Eng. 195(4), 207–213 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zienkiewicz, O., Zhu, J.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24, 337–357 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zienkiewicz, O., Zhu, J.: The super convergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Int. J. Numer. Methods Eng. 33(7), 1331–1364 (1992)

    Article  MATH  Google Scholar 

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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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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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