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A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

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Abstract

Due to the nonlocal property of the fractional derivative, the finite element analysis of fractional diffusion equation often leads to a dense and non-symmetric stiffness matrix, in contrast to the conventional finite element formulation with a particularly desirable symmetric and banded stiffness matrix structure for the typical diffusion equation. This work first proposes a finite element formulation that preserves the symmetry and banded stiffness matrix characteristics for the fractional diffusion equation. The key point of the proposed formulation is the symmetric weak form construction through introducing a fractional weight function. It turns out that the stiffness part of the present formulation is identical to its counterpart of the finite element method for the conventional diffusion equation and thus the stiffness matrix formulation becomes trivial. Meanwhile, the fractional derivative effect in the discrete formulation is completely transferred to the force vector, which is obviously much easier and efficient to compute than the dense fractional derivative stiffness matrix. Subsequently, it is further shown that for the general fractional advection–diffusion–reaction equation, the symmetric and banded structure can also be maintained for the diffusion stiffness matrix, although the total stiffness matrix is not symmetric in this case. More importantly, it is demonstrated that under certain conditions this symmetric diffusion stiffness matrix formulation is capable of producing very favorable numerical solutions in comparison with the conventional non-symmetric diffusion stiffness matrix finite element formulation. The effectiveness of the proposed methodology is illustrated through a series of numerical examples.

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Acknowledgements

The support of this work by the National Natural Science Foundation of China (11472233, 11772280) and the Natural Science Foundation of Fujian Province of China (2014J06001, 2014J01208) is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Civil Engineering, Xiamen University, Xiamen, 361005, Fujian, China

    Zeng Lin & Dongdong Wang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, China

    Zeng Lin

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  1. Zeng Lin
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  2. Dongdong Wang
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Correspondence to Dongdong Wang.

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Lin, Z., Wang, D. A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations. Comput Mech 62, 185–211 (2018). https://doi.org/10.1007/s00466-017-1492-2

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  • DOI: https://doi.org/10.1007/s00466-017-1492-2

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