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Abstract
In this paper, we study an efficient multigrid method to solve the semilinear interface problems. We first give an optimal finite element error estimate for the semilinear interface problems under a weak condition for the nonlinear term compared with the existing conclusions. Then next based on the finite element error estimate, we design a novel multigrid method for semilinear elliptic problems. The proposed multigrid method only requires to solve a linear interface problem in each level of the multilevel space sequence and a small-scale semilinear interface problem in a correction space. The involved linear interface problem can be solved efficiently by the multigrid iteration. The dimension of the correction space is small and fixed, which is independent from the fine spaces. Thus the computational time of the correction step is negligible compared with that of the linear interface problems in the fine spaces. On the whole, the efficiency of the presented multigrid method is nearly the same as that of the multigrid method for linear interface problems. Additionally, unlike the existing finite element error estimates and the multigrid methods for semilinear interface problems, which always require the bounded second order derivatives of the nonlinear terms, all the analysis in our paper only requires a Lipschitz continuous condition.
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