Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions
This work provides an efficient numerical method for solving nonlinear fractional PDEs, which is incremental as it combines existing techniques (TT-M, FE, θ-scheme) for a specific class of equations.
The authors developed a fast algorithm based on a time two-mesh finite element scheme for solving nonlinear space fractional Allen-Cahn equations, achieving reduced computational time while maintaining stability and accuracy, as demonstrated by numerical examples.
In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme, which aims at solving nonlinear problems quickly, is considered to numerically solve the nonlinear space fractional Allen-Cahn equations with smooth and non-smooth solutions. The implicit second-order $θ$ scheme containing both implicit Crank-Nicolson scheme and second-order backward difference method is applied to time direction, a fast TT-M method is used to increase the speed of calculation, and the FE method is developed to approximate the spacial direction. The TT-M FE algorithm includes the following main computing steps: firstly, a nonlinear implicit second-order $θ$ FE scheme on the time coarse mesh $τ_c$ is solved by a nonlinear iterative method; secondly, based on the chosen initial iterative value, a linearized FE system on time fine mesh $τ<τ_c$ is solved, where some useful coarse numerical solutions are found by the Lagrange's interpolation formula. The analysis for both stability and a priori error estimates are made in detail. Finally, three numerical examples with smooth and non-smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations, from which it is easy to find that the computing time can be saved.