Superconvergence analysis for a semilinear parabolic equation with BDF-3 finite element method

ABSTRACT

The main aim of this paper is to obtain superconvergence result for a semilinear parabolic equation with 3-step backward differential formula Galerkin finite element method. The time-discrete system is established to split the error into the temporal error and spatial error. The initial two steps of the temporal error is dealt with by a new way to ensure the third-order accuracy of the scheme. Some new tricks are utilized to get the spatial error in $H^1$-norm of order $O\left(h\right(h+{\tau }^{\frac{3}{2}}\left)\right)$ without the ratio between the spatial subdivision parameter h and the temporal step τ, which improves the corresponding results in the previous literature. The final superconvergence result is deduced by the above achievements and trigonometric inequality. Two numerical examples are provided to support the theoretical analysis.

COMMUNICATED BY:

• C. Bacuta

KEYWORDS:

• Superconvergence result
• semilinear parabolic equation
• BDF-3 FEM
• the temporal and the spatial errors

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65N15
• 65N30

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11671369) and Pingdingshan University High-level Talents Start-up Fund (PXY-BSQD-2019001).

Disclosure statement

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

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [No. 11671369] and Pingdingshan University High-level Talents Start-up Fund [PXY-BSQD-2019001].