Immersed finite element approximation for semi-linear parabolic interface problems combining with two-grid methods

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(h1/2) (or H=O(h1/4)).