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The local discontinuous Galerkin finite element method for a multiterm time-fractional initial-boundary value problem

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Abstract

In this paper, we study a multiterm time-fractional initial-boundary value problem, whose differential equation contains a sum of Caputo time fractional derivatives with orders in (0, 1). In general, the solution of this kind of problem exhibits a weak regularity at the initial time. Based on the L1 formula on non-uniform meshes for time discretization and the local discontinuous Galerkin (LDG) method for space discretization, fully discrete numerical schemes for one and two space dimensions are constructed. The stability and convergence of the schemes are analyzed. It is shown that the error bounds are \(\alpha _1\)-robust, that is, they remain valid as \(\alpha _1\rightarrow 1^-\), where \(\alpha _1\) is the biggest fractional order. Furthermore, a numerical experiment is given to verify the effectiveness of the current method.

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References

  1. Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the \(hp\)-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71(238), 455–478 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, H., Stynes, M.: Blow-up of error estimates in time-fractional initial-boundary value problems. IMA J. Numer. Anal. 41(2), 974–997 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)

    MATH  Google Scholar 

  4. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dong, B., Shu, C.-W.: Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems. SIAM J. Numer. Anal. 47, 3240–3268 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Du, Y., Liu, Y., Li, H., Fang, Z., He, S.: Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation. J. Comput. Phys. 344, 108–126 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Huang, C., An, N., Yu, X.: A local discontinuous Galerkin method for time-fractional diffusion equation with discontinuous coefficient. Appl. Numer. Math. 151, 367–379 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huang, C., An, N., Yu, X., Zhang, H.: A direct discontinuous Galerkin method for time-fractional diffusion equation with discontinuous diffusive coefficient. Complex Var. Elliptic Equ. 65(9), 1445–1461 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, C., Chen, H., Stynes, M.: An \(\alpha \)-robust finite element method for a multi-term time-fractional diffusion problem. J. Comput. Appl. Math. 389, 113334 (2021)

  10. Huang, C., Liu, X., Meng, X., Stynes, M.: Error analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem. Comput. Methods Appl. Math. 20(4), 815–825 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, C., Stynes, M., An, N.: Optimal \(L^\infty (L^2)\) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem. BIT Numer. Math. 58(3), 661–690 (2018)

    Article  MATH  Google Scholar 

  12. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Netherlands (2006)

    MATH  Google Scholar 

  13. Li, C.P., Cai, M.: Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia (2019)

    Book  Google Scholar 

  14. Li, C.P., Li, Z.Q., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin Method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput. 85(2), article 41 (2020)

  15. Li, C.P., Wang, Z.: The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: numerical analysis. Appl. Numer. Math. 140, 1–22 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, C.P., Wang, Z.: The discontinuous Galerkin finite element method for Caputo-type nonlinear conservation law. Math. Comput. Simulat. 169, 51–73 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luchko, Y.: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374(2), 538–548 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Meng, X.Y., Stynes, M.: Barrier function local and global analysis of an L1 finite element method for a multiterm time-fractional initial-boundary value problem. J. Sci. Comput. 84(1), article 5 (2020)

  19. Mustapha, K., McLean, W.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51(1), 491–515 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ren, J., Huang, C., An, N.: Direct discontinuous Galerkin method for solving nonlinear time fractional diffusion equation with weak singularity solution. Appl. Math. Lett. 102, 106111 (2020)

  21. Wang, H., Wang, S., Zhang, Q., Shu, C.-W.: Local discontinuous Galerkin methods with implicit-explicit time-marching for multidimensional convection-diffusion problems. ESAIM: M2AN 50(4), 1083–1105 (2016)

  22. Wei, L.L.: Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations. Numer. Algorithms 76(3), 695–707 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wei, L.L., He, Y.N.: Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl. Math. Model. 38(4), 1511–1522 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1–46 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Zeng, F., Zhang, Z., Karniadakis, G.E.: Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Comput. Methods Appl. Mech. Eng. 327, 478–502 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zaky, M.A.: A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput. Appl. Math. 37(3), 3525–3538 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, China

    Zhen Wang

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Wang, Z. The local discontinuous Galerkin finite element method for a multiterm time-fractional initial-boundary value problem. J. Appl. Math. Comput. 68, 4391–4413 (2022). https://doi.org/10.1007/s12190-021-01608-8

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  • DOI: https://doi.org/10.1007/s12190-021-01608-8

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