Unconditionally optimal time two-mesh mixed finite element algorithm for a nonlinear fourth-order distributed-order time fractional diffusion equation

Abstract

In this article, a fast second-order time two-mesh mixed finite element (TT-MMFE) algorithm is considered to numerically solve the nonlinear fourth-order distributed-order time fractional diffusion equation, where the generated formula for the fractional BDF2 and fractional trapezoidal rule through a free parameter θ (FBT-θ formula) combined with the composite trapezoid formula for the distributed-order derivative is used to discretize the time direction, and the MFE method is utilized to approximate the space direction, respectively. Further, to improve the computing efficiency of our numerical method, the TT-M algorithm is applied to deal with the nonlinear system. For the case of nonsmooth solution, the corrected TT-MMFE scheme by adding starting parts is designed. The existence and uniqueness of the numerical solution are proved in detail. Different from the traditional finite element analysis, the error between the exact solution and the finite element solution is split into temporal error and spatial error by introducing a so-called the error splitting technique, which helps us obtain the spatial error independent of the time step. Finally, numerical examples are given to validate that the TT-MMFE method can reduce computing time, and the corrected TT-MMFE scheme by adding starting parts can recover the convergence rate.