In order to analyze the supersonic and transonic panel flutter behaviors quantitatively and accurately, a fluid-structure coupling algorithm based on the finite element method (FEM) is proposed to study the two-dimensional panel flutter problem. First, the Von Kármán's large deformation is used to model the panel, and the high speed airflow is approached by the Euler equations. Then, the equation of panel is discretized spatially by the standard Galerkin FEM, and the equations of fluid are discretized by the characteristic-based split finite element method (CBS-FEM) with dual time stepping; thus, the numerical oscillation encountered frequently in the numerical simulation of flow field could be removed efficiently. Further, a staggered algorithm is used to transfer the information on the interface between the fluid and the structure. Finally, the aeroelastic behaviors of the panel in both the supersonic and transonic airflows are studied in details. And the results show that the system can present the flat and stable, simple harmonic oscillation, buckling, and inharmonic oscillation states at Mach 2, considering the effect of the pretightening force; at Mach 1.2, as the panel loses stability, the ensuing limit cycle oscillation is born; at Mach 0.8 and 0.9, positive and negative bucklings are the typical states of the panel as it loses its stability. Further, the transonic stability boundary is obtained and the transonic bucket is precisely captured. More, this algorithm can be applied to the numerical analysis of other complicated problems related to aeroelasticity.