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 problems. The partially penalized immersed finite element (PPIFE) methods are used for the spatial discretization to resolve discontinuity of the diffusion coefficient across the interface. The classical extrapolation method is adopted to treat the nonlinear term, which effectively avoids the complicated numerical calculation of the nonlinearity. Our error analysis 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 τ, we can unconditionally derive the optimal error estimates in both L2 norm and semi-H1 norm, while previous works always require the coupling condition of time step and space size. Numerical experiments are given to confirm the theoretical analysis.