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Convergence and quasi-optimality of an adaptive finite element method for semilinear elliptic problems on L 2 errors

In this paper, we prove the convergence and quasi-optimality of an adaptive finite element method for semilinear elliptic problems on errors by keeping sufficiently mildly graded meshes. Additional refinements are made to keep the meshes sufficiently mildly graded, but we find that it does not compromise the quasi-optimality of the adaptive finite element method presented in this paper. Numerical examples are provided to illustrate our theoretical results.