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

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...

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Bibliographic Details
Published in:	Physica. D Vol. 460; p. 134090
Main Authors:	Wen, Cao, Wang, Jinfeng, Liu, Yang, Li, Hong, Fang, Zhichao
Format:	Journal Article
Language:	English
Published:	Elsevier B.V 01.04.2024
Subjects:	Fourth-order distributed-order time fractional diffusion equation Mixed finite element method Time two-mesh algorithm Unconditional a priori error analysis Fourth-order distributed-order time fractional diffusion equation Mixed finite element method Time two-mesh algorithm Unconditional a priori error analysis
ISSN:	0167-2789, 1872-8022
Online Access:	Get full text
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Summary:	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. •A fast TT-MMFE algorithm is presented to solve the nonlinear distributed-order time fractional fourth-order diffusion model.•The error splitting technique is used to get unconditionally optimal L2-norm convergence.•Some numerical tests are carried out to show the computing efficiency of the fast TT-MMFE algorithm.•The corrected TT-MMFE algorithm by adding the correction terms is designed to help restore the convergence order.
ISSN:	0167-2789 1872-8022
DOI:	10.1016/j.physd.2024.134090