Two-grid scheme of expanded mixed finite element method for semilinear parabolic integro-differential equations

ABSTRACT

In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog scheme at the rest time levels. Next, we make use of the known coarse mesh solution and Taylor expansion to infer the solution on the fine mesh space. Thus, we only need to solve the nonlinear problem once on the coarse grid space of the two-grid scheme. Finally, a numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

COMMUNICATED BY:

• C. Bacuta

Keywords:

• Semilinear parabolic integro-differential equations
• mixed finite element method
• a priori error estimate
• two-grid
• Crank–Nicolson scheme

2010 Mathematics Subject Classifications:

• 49J20
• 65N30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The corresponding author is supported by Natural Science Foundation of China [11501473] and the Fundamental Research Funds of the Central Universities of China [2682016CX108]. The first author is supported by Technology Research Project of Jilin Provincial Department of Education [JJKH20190634KJ] and Youth Research and Innovation Team Development Project of Beihua University. The second author is supported by Graduate innovation program of Beihua University [2019001].