Two-grid methods of finite element solutions for semi-linear elliptic interface problems

In this paper, we present two efficient two-grid algorithms for solving two-dimensional semi-linear elliptic interface problems using finite element method. To linearize the finite element equations, the Newton iteration approach and correction technique are applied. The new two-grid schemes reduce the solution of the semi-linear interface problem on a fine grid to one linear interface equation on the same fine grid and an original interface problem on a much coarser grid. Therefore, the new schemes save total computational cost. Theoretical analysis shows that the two-grid methods maintain asymptotically optimal accuracy, and the numerical experiments presented confirm the theoretical results.