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ABSTRACT: In this paper, an adaptive finite element algorithm to control the error in $L^2$-norm is developed for second order elliptic equations. It complements the standard adaptive finite element method with a procedure to control the mesh size according to the a priori information on the second derivative of the solution. Optimal rate of convergence in $L^2$-norm is proved for convex domains in both two and three domains and for polygonal domains in two dimensions.