Superconvergence analysis for a nonlinear parabolic equation with a BDF finite element method

ABSTRACT

Superconvergence analysis for a nonlinear parabolic equation is studied with a linearized 2-step backward differential formula (BDF) Galerkin finite element method (FEM). The error between the exact solution and the numerical solution is split into two parts by a time-discrete system. The temporal error estimates in $H^1$-norm with order $O({\tau }^2)$ and in $H^2$-norm with order $O({\tau }^{3/2})$ are derived, respectively. The spatial error estimates are deduced unconditionally and the results help to bound the numerical solution in $L^{\mathrm{\infty }}$-norm. By some new way, the unconditional superclose property of $u^n$ in $H^1$-norm with order $O(h^2+{\tau }^2)$ is obtained. Two numerical examples show the validity of the theoretical analysis. Here, h is the subdivision parameter, and τ, time step size.

Keywords:

• Nonlinear parabolic equation
• BDF Galerkin FEM
• temporal error
• spatial error
• unconditional superclose result

2010 Mathematics Subject Classifications:

• 65N15
• 65N30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (No. 11671369), the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 20A110030), the Doctoral Starting Foundation of Pingdingshan University (No. PXY-BSQD-2019001) and the University Cultivation Foundation of Pingdingshan (No. PXY-PYJJ-2019006).