In this paper, we will focus on adaptive meshing and re-meshing. We present an original approach, based on Quadtree and Octree, to construct the initial mesh and refine it using mixed-elements. We propose a fast algorithm using a determined set of Patterns to handle transitions between fine and coarse regions, and to closely approximate surface boundaries. An optimized structure for storing edges permits to efficiently manage neighbor information, dramatically reducing the balancing time. Results, both in 2D and 3D, exhibit the multiple possibilities for local refinements. Comparisons in terms of accuracy, number of elements in the resulting mesh, and computation time, clearly position our method as a competitive alternative to generate a conform adapted mesh, which can be used with standard numerical methods.