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Fractional SUPG finite element formulation for multi-dimensional fractional advection diffusion equations

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Abstract

Multi-dimensional advection diffusion equations involving fractional diffusion flux are studied with finite element formulation. It has been demonstrated that the Galerkin finite element formulation may suffer spatial instability and nodal oscillations due to the fractional diffusion flux in addition to the advection term. To resolve this issue, a fractional streamline upwind Petrov-Galerkin finite element formulation is developed. In this formulation, an artificial viscosity is added to the test function to eliminate the spatial oscillations. By taking into account both the fractional diffusion flux and the advection term, a decomposition algorithm is proposed to formulate the artificial viscosity to avoid the cross-wind diffusion effect for multi-dimensional problems. Numerical examples with regular/irregular domains discretized by uniform/nonuniform meshes for both steady state and time-dependent fractional advection diffusion equations are presented to thoroughly demonstrate the effectiveness of the proposed methodology.

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References

  1. Benson DA, Schumer R, Meerschaert MM, Wheatcraft SW (2001) Fractional dispersion, Lev motion, and the MADE tracer tests. Transp Porous Media 42:211–240

    Article  MathSciNet  Google Scholar 

  2. Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier-stokes equations. Comput Methods Appl Mech Eng 32(1):199–259

    Article  MathSciNet  Google Scholar 

  3. Caputo M, Mainardi F (1971) Linear models of dissipation in an elastic solids. Rivista Del Nuevo Cimento (Ser II) 1:161–198

    Article  Google Scholar 

  4. Chen W, Sun HG, Zhang XD, Korosak D (2010) Anomalous diffusion modeling by fractal and fractional derivatives. Comput Math Appl 59:1754–1758

    Article  MathSciNet  Google Scholar 

  5. Du N, Guo X, Wang H (2020) Fast upwind and Eulerian-Lagrangian control volume schemes for time-dependent directional space-fractional advection-dispersion equations. J Comput Phys 405:109127

    Article  MathSciNet  Google Scholar 

  6. Duo S, Wang H (2019) A fractional phase-field model using an infinitesimal generator of alpha stable levy process. J Comput Phys 384:253–269

    Article  MathSciNet  Google Scholar 

  7. Ervian VJ, Roop JP (2005) Variational formulation for the stationary fractional advection dispersion equation. Numer. Meth. PDE. 22:558–576

    Article  MathSciNet  Google Scholar 

  8. Ervin VJ, Roop JP (2007) Variational solution of fractional advectoin dispersion equations on bounded domains in \(\mathbb{R}^d\). Numer Meth PDE 23:256–281

    Article  Google Scholar 

  9. Fan W, Liu F (2018) A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain. Appl Math Lett

  10. Hughes TJR, Brooks A (1982) A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: application to the streamline-upwind procedure. Finite elements in fluids 4:47–65

    Google Scholar 

  11. Hughes TJR, Feijoo GR, Mazzei L, Quincy J-B (1998) The variational multiscale method paradigm for computational mechanics. Comput Methods Appl Mech Eng 166(1):3–24

    Article  MathSciNet  Google Scholar 

  12. Hughes TJR, Franca LP, Mallet M (1987) A new finite element formulation for computational fluid dynamics: Vi convergence analysis of the generalized supg formulation for linear time-dependent multidimensional advection-diffusion systems. Comput Methods Appl Mech Eng 63(1):97–112

    Article  Google Scholar 

  13. Hughes TJR, Mallet M, Akira M (1986) A new finite element formulation for computational fluid dynamics: Ii. beyond supg. Comput Methods Appl Mech Eng 54(3):341–355

    Article  MathSciNet  Google Scholar 

  14. Jin B, Lazarov R, Zhou Z (2016) A petrov-galerkin finite element method for fractional convection-diffusion equations. SIAM J Numer Anal 54(1):481–503

    Article  MathSciNet  Google Scholar 

  15. Karniadakis GEM, Hesthaven JS, Podlubny I (2015) Special issue on “Fractional PDEs: Theory, Numerics, and Applications”. J Comput Phys 293:1–3

    Article  MathSciNet  Google Scholar 

  16. Li Z, Wang H, Yang D (2017) A space-time fractional phase-field model with tunable sharpness and decay behavior and its efficien numerical simulation. J Comput Phys 347:20–38

    Article  MathSciNet  Google Scholar 

  17. Lian YP, Ying YP, Tang SQ, Lin S, Wagner GJ, Liu WK (2016) A Petrov–Galerkin finite element method for the fractional advection-diffusion equation. Comput Methods Appl Mech Eng 309:388–410

    Article  MathSciNet  Google Scholar 

  18. Lin Z, Wang D, Qi D, Deng L (2020) A Petrov-Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations. Comput Mech 66:323–350

  19. Lin Z, Wang DD (2017) A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations. Comput Mech

  20. Lischke A, Pang G, Gulian M, Song F, Glusa C, Zheng X, Mao Z, Cai M, Meerschaert MM, Ainsworth M, Em Karniadakis G (2020) What is the fractional Laplacian ? A comparative review with new results. J Comput Phys 404:109009

    Article  MathSciNet  Google Scholar 

  21. Luan S, Lian Y, Ying Y, Tang S, Wagner GJ, Liu WK (2017) An enriched finite element method to fractional advection-diffusion equation. Comput Mech 60:181–201

    Article  MathSciNet  Google Scholar 

  22. Meerschaert MM, Scheffler HP, Tadjeran C (2006) Finite difference methods for two-dimensional fractional dispersion equation. J Comput Phys 211:249–261

    Article  MathSciNet  Google Scholar 

  23. Meerschaert MM, Tadjeran C (1999) Fractional Brownian motion via fractional Laplacian. Stat Probab Lett 44:107–108

    Article  MathSciNet  Google Scholar 

  24. Meerschaert MM, Tadjeran C (2004) Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math 172:65–77

    Article  MathSciNet  Google Scholar 

  25. Pang G, Chen W, Fu Z (2015) Space-fractional advection-dispersion equations by the Kansa method. J Comput Phys 293:280–296

    Article  MathSciNet  Google Scholar 

  26. Pang G, Chen W, Sze KY (2016) A comparative study of finite element and finite difference methods for two-dimensional space-fractional advection-dispersion equation. Adv Appl Math Mech 8:166–186

    Article  MathSciNet  Google Scholar 

  27. Roop JP (2006) Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \(\mathbb{R}^d\). J Comput Appl Math 193:243–268

    Article  MathSciNet  Google Scholar 

  28. Sun HG, Zhang Y, Chen W, Reeves DM (2014) Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J Contam Hydrol 157:47–58

    Article  Google Scholar 

  29. Sun HG, Zhang Y, Baleanu D, Chen W, Chen YQ (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213–231

    Article  Google Scholar 

  30. Wang H, Basu TS (2012) A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J Sci Comput 34(5):A2444–A2458

    Article  MathSciNet  Google Scholar 

  31. Wang H, Yang DP, Zhu SF (2015) A petrov-galerkin finite element method for variable-coefficient fractional diffusion equations. Comput Methods Appl Mech Eng 290:45–56

    Article  MathSciNet  Google Scholar 

  32. Wang Y, Yan Y, Hu Y (2019) Numerical methods for solving space fractional partial differential equations using hadamard finite-part integral approach. Commun Appl Math Comput 1:505–523

    Article  MathSciNet  Google Scholar 

  33. Yang Z, Liu Y, Nie F, Turner I (2020) An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains. J Comput Phys 408:109284

    Article  MathSciNet  Google Scholar 

  34. Zhao X, Hu X, Cai W, Karniadakis GE (2017) Adaptive finite element method for fractional differential equations using hierarchial matrics. Comput Methods Appl Mech Eng 325:56–76

    Article  Google Scholar 

  35. Zhao Y, Bu W, Zhao X, Tang Y (2017) Galerkin finite element method for two-dimensional space and time fractional Bloch-Torrey equation. J Comput Phys 350:117–135

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Y. P. Lian acknowledges the support by the National Natural Science Foundation of China (NSFC) under grant No. 11972086.

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Authors and Affiliations

  1. Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing, 100081, China

    Mingji Chen & Yanping Lian

  2. Department of Civil and Systems Engineering, Johns Hopkins University, Baltimore, MD, 21218, USA

    Shengzhi Luan

  3. State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, 100081, China

    Yanping Lian

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  1. Mingji Chen
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  2. Shengzhi Luan
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  3. Yanping Lian
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Correspondence to Shengzhi Luan or Yanping Lian.

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Chen, M., Luan, S. & Lian, Y. Fractional SUPG finite element formulation for multi-dimensional fractional advection diffusion equations. Comput Mech 67, 601–617 (2021). https://doi.org/10.1007/s00466-020-01951-w

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  • DOI: https://doi.org/10.1007/s00466-020-01951-w

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