Communications in Mathematical Sciences
Volume 21 (2023)
Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations
Pages: 775 – 794
In this paper we consider a linearized variable-time-step two-step backward differentiation formula (BDF2) scheme for solving nonlinear parabolic equations. The scheme is constructed by using the variable time-step BDF2 for the linear term and a Newton linearized method for the nonlinear term in time combining with a Galerkin finite element method (FEM) in space. We prove the unconditionally optimal error estimate of the proposed scheme under mild restrictions on the ratio of adjacent time-steps, i.e. the ratio less than $4.8645$, and on the maximum time step. The proof involves the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and the error splitting approach. In addition, our analysis also shows that the first level solution obtained by BDF1 (i.e., backward Euler scheme) does not cause the loss of global accuracy of second order. Numerical examples are provided to demonstrate our theoretical results.
nonlinear parabolic equations, variable time-step BDF2, orthogonal convolution kernels, stability and convergence, error splitting approach
2010 Mathematics Subject Classification
Jiwei Zhang is supported in the NSFC under grants nos. 12171376 and 2020-JCJQ-ZD-029, and by the Fundamental Research Funds for Central Universities, grant no. 2042021kf0050.
Received 22 January 2022
Received revised 23 July 2022
Accepted 5 August 2022
Published 27 February 2023