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Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model

Abstract

In this article, a time two-mesh (TT-M) algorithm combined with the H1-Galerkin mixed finite element (FE) method is introduced to numerically solve the nonlinear distributed-order sub-diffusion model, which is faster than the H1-Galerkin mixed FE method. The Crank-Nicolson scheme with TT-M algorithm is used to discretize the temporal direction at time \(t_{n+\frac {1}{2}}\), the FBN-𝜃 formula is developed to approximate the distributed-order derivative, and the H1-Galerkin mixed FE method is used to approximate the spatial direction. TT-M mixed element algorithm mainly covers three steps: first, the mixed finite element solution of the nonlinear coupled system on the time coarse mesh ΔtC is calculated; next, based on the numerical solution obtained in the first step, the numerical solution of the nonlinear coupled system on time fine mesh ΔtF is obtained by using Lagrange’s interpolation formula; finally, the numerical solution of the linearized system on time fine mesh ΔtF is solved by using the results in the second step. The existence and uniqueness of the solution for our numerical scheme are shown. Moreover, the stability and a priori error estimate are analyzed in detail. Furthermore, numerical examples with smooth and nonsmooth solutions are given to validate our method.

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Acknowledgments

The authors are grateful to the referees and editor for their valuable comments and good suggestions which greatly improved the presentation of the paper.

Funding

This work is supported by the National Natural Science Foundation of China (12061053, 11661058, 11761053), Natural Science Foundation of Inner Mongolia (2020MS01003), and program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-17-A07).

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Correspondence to Yang Liu.

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Wen, C., Liu, Y., Yin, B. et al. Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model. Numer Algor 88, 523–553 (2021). https://doi.org/10.1007/s11075-020-01048-8

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Keywords

  • Time two-mesh algorithm
  • H 1-Galerkin mixed finite element method
  • Crank-Nicolson formula
  • Nonlinear distributed-order sub-diffusion model
  • Stability and a priori error analysis