Convergence of a linearized second-order BDF-FEM for nonlinear parabolic interface problems

We present an almost optimal error estimate for the finite element solution of a nonlinear parabolic interface problem, where the coefficient depends on the unknown variable and is discontinuous along an interface inside the computational domain. A linearized second-order backward difference formula is used for the time discretization, and piecewise linear interpolation is used to approximate the interface. We do not assume Lipschitz continuity of the nonlinear coefficient.