A Petrov–Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations

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Meshfree methods with arbitrary order smooth approximation are very attractive for accurate numerical modeling of fractional differential equations, especially for multi-dimensional problems. However, the non-local property of fractional derivatives poses considerable difficulty and complexity for the numerical simulations of fractional differential equations and this issue becomes much more severe for meshfree methods due to the rational nature of their shape functions. In order to resolve this issue, a new weak formulation regarding multi-dimensional Riemann–Liouville fractional diffusion equations is introduced through unequally splitting the original fractional derivative of the governing equation into a fractional derivative for the weight function and an integer derivative for the trial function. Accordingly, a Petrov–Galerkin finite element-meshfree method is developed, where smooth reproducing kernel meshfree shape functions are adopted for the trial function approximation to enhance the solution accuracy, and the discretization of weight function is realized by the explicit finite element shape functions with an analytical fractional derivative evaluation to further reduce the computational complexity and improve efficiency. The proposed method enables a direct and efficient employment of meshfree approximation, and also eliminates the undesirable singular integration problem arising in the fractional derivative computation of meshfree shape functions. A nonlinear extension of the proposed method to the fractional Allen–Cahn equation is presented as well. The effectiveness of the proposed methodology is consistently demonstrated by numerical results.