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

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.