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Directors:
Randolph E. Bank
Philip E. Gill
Michael Holst

In this talk, we consider solving semilinear PDEs with discontinuous diffusion coefficients by a two-grid algorithm. The algorithm consists of a coarse solver on the original nonlinear problem, and a single linear Newton update. Under the assumption that the nonlinear function is monotone, we derive the a priori $L^{\infty}$ bounds of the continuous solution, and $L^{\infty}$ bounds on the discrete solutions with additional angle condition on the triangulation. With the help of these a priori $L^{\infty}$ bounds, we derive quasi-optimal error estimate. We also derive the $L^2$ error estimate via duality argument. Finally, we give the error estimate on the numerical solution generated by the two-grid algorithm. Numerical results justify our theoretical conclusions.