Variational formulation for fractional inhomogeneous boundary value problems

Abstract

The steady state fractional convection diffusion equation with inhomogeneous Dirichlet boundary is considered. By utilizing standard boundary shifting trick, a homogeneous boundary problem is derived with a singular source term which does not belong to \(L^2\) anymore. The variational formulation of such problem is established, based on which the finite element approximation scheme is developed. Inf-sup conditions for both continuous case and discrete case are demonstrated thus the corresponding well-posedness is verified. Furthermore, rigorous regularity analysis for the solutions of both original equation and dual problem is carried out, based on which the error estimates for the finite element approximation are derived. Numerical results are presented to illustrate the theoretical analysis.

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Acknowledgements

We would like to thank the anonymous referees for many constructive comments and suggestions which led to an improved presentation of this paper.

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Correspondence to Zhoushun Zheng.

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This work was supported by the National Key Research and Development Program of China (Grant Nos. 2017YFB0701700, 2017YFB0305601).

Communicated by Mihaly Kovacs.

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Fu, T., Zheng, Z. & Duan, B. Variational formulation for fractional inhomogeneous boundary value problems. Bit Numer Math (2020). https://doi.org/10.1007/s10543-020-00812-5

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Keywords

  • Fractional calculus
  • Variational formulation
  • Inhomogeneous boundary value
  • Well-posedness

Mathematics Subject Classification

  • 26A33
  • 65N12
  • 65N30