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Journal of Scientific Computing

, Volume 77, Issue 3, pp 1468–1489 | Cite as

Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions

  • Sheng Chen
  • Jie Shen
Article
  • 56 Downloads

Abstract

Usual spectral methods are very effective for problems with smooth solutions, but their accuracy will be severely limited if solution of the underlying problems exhibits singular behavior. We develop in this paper enriched spectral-Galerkin methods (ESG) to deal with a class of problems for which the form of leading singular solutions can be determined. Several strategies are developed to overcome the ill conditioning due to the addition of singular functions as basis functions, and to efficiently solve the approximate solution in the enriched space. We validate ESG by solving a variety of elliptic problems with weakly singular solutions.

Keywords

Weakly singular solution Spectral-Galerkin method Enriched space Jacobi polynomials Error estimate 

Mathematics Subject Classification

65N35 41A10 41A30 41A99 

References

  1. 1.
    Adcock, B., Huybrechs, D.: Frames and numerical approximation (2016). arXiv:1612.04464 [math.NA]
  2. 2.
    Auteri, F., Parolini, N., Quartapelle, L.: Numerical investigation on the stability of singular driven cavity flow. J. Comput. Phys. 183(1), 1–25 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Babuška, I., Banerjee, U.: Stable generalized finite element method (SGFEM). Comput. Method Appl. M. 201, 91–111 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Babuška, I., Miller, A.: The post–processing approach in the finite element method–Part 2: The calculation of stress intensity factors. Int. J. Numer. Meth. Eng. 20(6), 1111–1129 (1984)CrossRefGoogle Scholar
  5. 5.
    Björck, A., Paige, C.C.: Loss and recapture of orthogonality in the modified gram-schmidt algorithm. SIAM J. Matrix Anal. Appl. 13(1), 176–190 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Botella, O., Peyret, R.: Computing singular solutions of the naviercstokes equations with the chebyshev collocation method. Int. J. Numer. Methods Fluids 36, 125 (2001)CrossRefGoogle Scholar
  7. 7.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2001)zbMATHGoogle Scholar
  8. 8.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional laplacian. Comm. Partial Differ. Eq. 32(8), 1245–1260 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)zbMATHGoogle Scholar
  10. 10.
    Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some non-local semilinear equations. Commun. Part. Differ. Eq. 36(8), 1353–1384 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, S., Shen, J., Wang, L.L.: Generalized jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Math, vol. 2004. Springer, Berlin (2010)Google Scholar
  13. 13.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fries, T.P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84(3), 253–304 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gorenflo, R., Mainardi, F.: Fractional oscillations and Mittag-Leffler functions. Citeseer, Prineton (1996)zbMATHGoogle Scholar
  16. 16.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. Number 26 in CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1977)Google Scholar
  17. 17.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, vol. 69. SIAM, Philadelphia (2011)CrossRefGoogle Scholar
  18. 18.
    Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, 298628 (2011).  https://doi.org/10.1155/2011/298628 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  20. 20.
    Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, Z.C., Lu, T.T.: Singularities and treatments of elliptic boundary value problems. Math. Comput. Mod. 31(8), 97–145 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, Z.C., Lu, T.T., Hu, H.Y., Cheng, A.H.: Particular solutions of laplace’s equations on polygons and new models involving mild singularities. Eng. Anal. Bound. Elem. 29(1), 59–75 (2005)CrossRefGoogle Scholar
  23. 23.
    Mao, Z., Chen, S., Shen, J.: Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Appl. Numer. Math. 106, 165–181 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mao, Z., Karniadakis, G.E.: A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J. Numer. Anal. 56(1), 24–49 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Marin, L., Lesnic, D., Mantič, V.: Treatment of singularities in helmholtz-type equations using the boundary element method. J. Sound Vib. 278(1), 39–62 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Morin, P., Nochetto, R., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (2002)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nochetto, R.H., Otárola, E., Salgado, A.J.: A pde approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733–791 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Podlubny, I.: Fractional Differential Equations, volume 198 of Mathematics in Science and Engineering. Academic Press Inc., San Diego, CA, (1999). An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applicationsGoogle Scholar
  29. 29.
    Samko, S.G., Kilbas, A.A., Maričev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publ, Philadelphia (1993)Google Scholar
  30. 30.
    Schultz, W.W., Lee, N.Y., Boyd, J.P.: Chebyshev pseudospeetral method of viscous flows with corner singularities. J. Sci. Comput. 4, 1–24 (1989)CrossRefGoogle Scholar
  31. 31.
    Schumack, M.R., Schultz, W.W., Boyd, J.P.: Spectral method solution of the stokes equations on nonstaggered grids. J. Comput. Phys. 94(1), 30–58 (1991)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Shen, J.: Efficient spectral-Galerkin method I. Direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489–1505 (1994)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Shen, J.: Efficient spectral-Galerkin methods III. Polar and cylindrical geometries. SIAM J. Sci. Comput. 18, 1583–1604 (1997)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Series in Computational Mathematics, vol. 41. Springer, Berlin (2011)Google Scholar
  35. 35.
    Shen, J., Wang, Y.: Müntz–Galerkin methods and applications to mixed Dirichlet–Neumann boundary value problems. SIAM J. Sci. Comput. 38(4), A2357–A2381 (2016)CrossRefGoogle Scholar
  36. 36.
    Sherman, A.H.: On newton-iterative methods for the solution of systems of nonlinear equations. SIAM J. Numer. Anal. 15(4), 755–771 (1978)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall Series in Automatic Computation. Prentice-Hall Inc, Englewood Cliffs (1973)zbMATHGoogle Scholar
  38. 38.
    Szego, G.: Orthogonal Polynomials, vol. 23 of Amer. In: Math. Soc. Colloq. Publ., Amer. Math. Soc., Providence, RI (1975)Google Scholar
  39. 39.
    Zayernouri, M., Karniadakis, G.E.: Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys. 257(Part A), 460–480 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouPeople’s Republic of China
  2. 2.Beijing Computational Science Research CenterBeijingPeople’s Republic of China
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA
  4. 4.School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific ComputingXiamen UniversityXiamenPeople’s Republic of China

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