Linearly implicit BDF methods for nonlinear parabolic interface problems

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We analyze linearly implicit BDF methods for the time discretization of a nonlinear parabolic interface problem, where the computational domain is divided into two subdomains by an interface, and the nonlinear diffusion coefficient is discontinuous across the interface. We prove optimal-order error estimates without assuming any growth conditions on the nonlinear diffusion coefficient and without restriction on the stepsize. Due to the existence of the interface and the lack of global Lipschitz continuity of the diffusion coefficient, we use a special type of test functions to analyze high-order \(A(\alpha )\)-stable BDF methods. Such test functions avoid any interface terms upon integration by parts and are used to derive error estimates in the piecewise \(H^2\) norm.