Optimal convergence analysis of a linearized second-order BDF-PPIFE method for semi-linear parabolic interface problems

The article proposes and analyzes the optimal error estimates of a second-order backward difference formula (BDF2) numerical scheme for the semi-linear parabolic interface prob-lems. The partially penalized immersed finite element (PPIFE) methods are used for the spatial discretization to resolve discontinuity of the diffusion coefficient across the inter-face. The classical extrapolation method is adopted to treat the nonlinear term, which ef-fectively avoids the complicated numerical calculation of the nonlinearity. Our error anal-ysis is based on the corresponding time-discrete system, which neatly splits the error into two parts: the temporal discretization error and the spatial discretization error. Since the spatial discretization error is independent of time step size Tau, we can unconditionally de-rive the optimal error estimates in both L 2 norm and semi -H 1 norm, while previous works always require the coupling condition of time step and space size. Numerical experiments are given to confirm the theoretical analysis.(c) 2022 Elsevier Inc. All rights reserved.