In this paper, Fluidity-Atmosphere, representative of a three-dimensional (3D) non-hydrostatic Galerkin compressible atmospheric dynamic framework, is generated to resolve large-scale and small-scale phenomena simultaneously. This achievement is facilitated by the use of non-hydrostatic equations and the adoption of a flexible 3D dynamically adaptive mesh where the mesh is denser in areas with higher gradients of variable solutions and relatively sparser in the rest of the domain while maintaining promising accuracy and reducing computational resource requirements. The dynamic core is formulated based on anisotropic tetrahedral meshes in both the horizontal and vertical directions. The performance of the adaptive mesh techniques in Fluidity-Atmosphere is evaluated by simulating the formation and propagation of a non-hydrostatic mountain wave. The 2D anisotropic adaptive mesh shows that the numerical solution is in good agreement with the analytic solution. The variation in the horizontal and vertical resolutions has a strong impact on the smoothness of the results and maintains convergence even at high resolutions. When the simulation is extended to 3D, Fluidity-Atmosphere shows stable and symmetric results in the benchmark test cases. The flows over a bell-shaped mountain are resolved quite smoothly. For steep mountains, Fluidity-Atmosphere performs very well, which shows the potential of using 3D adaptive meshes in atmospheric modeling. Finally, as an alternative cut-cell mesh in Fluidity-Atmosphere, the anisotropic adaptive mesh coupled with the Galerkin method provides an alternative accurate representation of terrain-induced flow.