A new adaptive remeshing approach for unstructured meshes, which includes the error indicator, global and local mesh regeneration techniques, has been developed in this paper. In this approach, nodes are first distributed according to the remeshing parameters, and those nodes are connected into a complete mesh. The concepts of side-based and vertex-based fronts are introduced to achieve the triangulation. According to the CPU time and the versatility, the vertex-based triangulation technique is proved to be more efficient. By using vertexbased triangulation approach, a local remeshing method, which regenerates only some regions of the flow domain, is presented. To demonstrate the reliability and availability of the proposed procedure, several compressible flow problems are investigated. The regular/stretched triangles and the mixed type of triangular and quadrilateral stretched elements are used. In this work, the Euler equations are solved by the multi-step Runge-Kutta Galerkin finite element methods. From the numerical results, the approaches, which employ the directionally stretched elements, are effective and suitable for treating the flow problems with one-dimensional features. The development of the local remeshing algorithm for unsteady flows is worthwhile and important.