Adaptive mesh refinement techniques are used in order to decrease the computational cost associated with the numerical solution of Partial Differential Equations. In this work, the refined mesh is represented by a graph data structure. More precisely. this scheme follows the Autonomous Leaves Graph concepts. The objective is to construct an adaptive mesh refinement with lower cost than tree-based schemes. Moreover, the Autonomous Leaves Graph was initially proposed with the Finite Volume Method and a Modified Hilbert Curve was used for the total-ordering of the control volumes. This work proposes to integrate the Autonomous Leaves Graph and the Finite Element Method as well as to adapt the Modified Hilbert Curve for this scheme. Furthermore, a non-conforming
-adaptive strategy is implemented. This approach is applied in the solution of the Poisson equation problem and the experimental results are discussed.