Abstract: Based on the successful performance of the reduced element for initial value problems, the reduced element for general boundary value problems is proposed without the needs for super-convergence calculations, structural meshes and nodal accuracy improvement. An adaptive finite element algorithm with the solution of the reduced element and a built-in maximum norm error estimator as the final results is subsequently proposed. This paper gives a brief report to this research progress and provides preliminary numerical examples in one- and two-dimensional boundary value problems to exhibit the feasibility, effectiveness and reliability of the proposed algorithm.