A two-grid immersed finite element method with the Crank-Nicolson time scheme for semilinear parabolic interface problems

Abstract

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 L2 norm and semi-H1 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 L2 norm (or semi-H1 norm) when the mesh size satisfies H=O(h3/2) (or H=O(h3)).