The paper is concerned with the grid refinement/coarsening/alignment technique for the adaptive solution of the 3D compressible flow. The necessary condition for the properties of the tetrahedrization on which the discretization error is below the prescribed tolerance is formulated. The interpolation error in the density and in the Mach number for the inviscid and viscous flow, respectively, is used to control this necessary condition. The original smoothing procedure for the generally discontinuous approximate solution is proposed in terms of edge based Hesse matrices. This allows to define the concept of the optimal mesh and to adopt the standard anisotropic mesh adaptation technique for the equidistribution of the interpolation error function. Geometrically interpreted, the optimal mesh is almost equilateral tetrahedrization with respect to the solution dependent Riemann norm of edges. For its construction, the iterative algorithm is proposed. The theoretical considerations are completed by the numerical example of the adaptive solution of the 3D channel flow.
