  In this article we consider two-grid finite element methods for solving
semilinear interface problems in d space dimensions, for d=2 or d=3. We first
describe in some detail the target problem class with discontinuous diffusion
coefficients, which includes problems containing sub-critical, critical, and
supercritical nonlinearities. We then establish basic quasi-optimal a priori
error estimate for Galerkin approximations. In the critical and subcritical
cases, we follow our recent approach to controling the nonlinearity using only
pointwise control of the continuous solution and a local Lipschitz property,
rather than through pointwise control of the discrete solution; this eliminates
the requirement that the discrete solution satisfy a discrete form of the
maximum principle, hence eliminating the need for restrictive angle conditions
in the underlying mesh. The supercritical case continues to require such mesh
conditions in order to control the nonlinearity. We then design a two-grid
algorithm consisting of a coarse grid solver for the original nonlinear
problem, and a fine grid solver for a linearized problem. We analyze the
quality of approximations generated by the algorithm, and show that the coarse
grid may be taken to have much larger elements than the fine grid, and yet one
can still obtain approximation quality that is asymptotically as good as
solving the original nonlinear problem on the fine mesh. The algorithm we
describe, and its analysis in this article, combines four sets of tools: the
work of Xu and Zhou on two-grid algorithms for semilinear problems; the recent
results for linear interface problems due to Li, Melenk, Wohlmuth, and Zou;
recent work on the Poisson-Boltzmann equation; and recent work on a priori
estimates for semilinear problems.
