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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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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Communicated by Mihaly Kovacs.
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This work was supported by the National Key Research and Development Program of China (Grant Nos. 2017YFB0701700, 2017YFB0305601).
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Fu, T., Zheng, Z. & Duan, B. Variational formulation for fractional inhomogeneous boundary value problems. Bit Numer Math 60, 1203–1219 (2020). https://doi.org/10.1007/s10543-020-00812-5
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DOI: https://doi.org/10.1007/s10543-020-00812-5