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In this paper, we propose and analyze a new two-grid partially penalized immersed finite element method for solving the semilinear parabolic interface problems with meshes independent of the coefficient discontinuity. Based on the corresponding time-discrete system, we can unconditionally derive the optimal error estimates in both the L 2 norm and semi- H 1 norm, while previous works always require the coupling condition of time step and space size (e.g. condition τ = O (H)). Then, we design a two-grid algorithm based on Newton iteration to deal with nonlinear source term. It is shown, both theoretically and numerically, that the algorithm can achieve asymptotically optimal approximation in L 2 norm (or semi- H 1 norm) when the mesh size satisfies H = O (h 3 / 2) (or H = O (h 3)). [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze the two-grid immersed finite element methods for semi-linear parabolic interface problems with discontinuous diffusion coefficients. The immersed finite element methods are used for spatial discretization where the meshes are not aligned with the interface. Optimal error estimates have been derived for both spatially semi-discrete schemes and fully discrete schemes. The two-grid algorithms based on the Newton methods are adopted to treat the nonlinear term. It is theoretically and numerically illustrated that the two-grid immersed finite element methods can achieve optimal convergence order when the coarse mesh satisfies H = O (h 1 / 2) (or H = O (h 1 / 4)). [ABSTRACT FROM AUTHOR]