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 $$L^2$$ 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.

中文翻译：

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分数非齐次边值问题的变分公式

考虑了具有非均匀狄利克雷边界的稳态分数对流扩散方程。通过使用标准的边界转移技巧，一个齐次边界问题被推导出一个不再属于 $$L^2$$L 2 的奇异源项。建立了此类问题的变分公式，并在此基础上开发了有限元近似方案。证明了连续情况和离散情况的 Inf-sup 条件，从而验证了相应的适定性。此外，对原方程和对偶问题的解进行严格的正则性分析，在此基础上推导出有限元近似的误差估计。给出了数值结果来说明理论分析。