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.

用有限元公式研究了包含分数扩散通量的多维对流扩散方程。已经证明，由于对流项之外，由于分数扩散通量，Galerkin有限元公式可能会遭受空间不稳定性和节点振荡。为解决此问题，开发了分流式上风Petrov-Galerkin有限元公式。在这种配方中，将人工粘度添加到测试功能中以消除空间振荡。通过同时考虑分数扩散通量和对流项，提出了一种分解算法来计算人工黏度，从而避免了多维问题的侧风扩散效应。