Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 323–341

# Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order

• Leilei Wei
• Lijie Liu
• Huixia Sun
Original Research

## Abstract

In this paper, a numerical method is proposed for solving distributed order diffusion equation, which arises in the mathematical modeling of ultra-slow diffusion processes observed in some physical problems, whose solution decays logarithmically as the time t tends to infinity. Based on local discontinuous Galerkin method in space, we develop a fully discrete scheme and prove that the scheme is unconditionally stable and convergent with the order $$O(h^{k+1}+\Delta t+\Delta \alpha ^2)$$, where $$h, \Delta t$$,$$\Delta \alpha$$ and k are the step size in space, time, distributed order and the degree of piecewise polynomials, respectively. Extensive numerical examples are carried out to illustrate the effectiveness of the numerical schemes.

## Keywords

Fractional diffusion equation Stabilized finite element method Stability Error estimate

## Mathematics Subject Classification

65M12 65M06 35S10

## Notes

### Acknowledgements

This work is supported by the High-Level Personal Foundation of Henan University of Technology (2013BS041), Plan For Scientific Innovation Talent of Henan University of Technology (2013CXRC12), the National Natural Science Foundation of China (11426090, 11461072), and China Postdoctoral Science Foundation funded Project (2015M572114).

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© Korean Society for Computational and Applied Mathematics 2018

## Authors and Affiliations

• Leilei Wei
• 1
• Lijie Liu
• 1
• Huixia Sun
• 1
1. 1.College of ScienceHenan University of TechnologyZhengzhouPeople’s Republic of China