We present an iterative algorithm for mathematical modeling of an elastic deformation process in a fluid-saturated fractured-porous medium. A three-dimensional multi-physics problem describes the coupled isothermal processes of the solid elastic deformation and slightly compressible fluid flow under external pressure. Mathematical models of these processes are connected via interface conditions for the pressure and density fields on the surface of a fractured-porous medium. For solving the multi-physics problem, a special multiscale procedure was developed. We use a heterogeneous multiscale finite element discretization on coarse polyhedral grids for the solid elastic deformation problem. Multiscale shape functions are constructed using special interface conditions for a hydrodynamic pressure on the surface of pores. We apply a discontinuous Galerkin method and a stabilized finite element discretization on fine tetrahedral grids for solving the hydrodynamics problem in fluid-saturated pores. In this case, we can realize an effective parallel procedure for solving the multi-physics problem. In each pore, hydrodynamics problems can be solved in parallel and independently. Verifications of the computational schemes are presented. We consider three-dimensional media with a different volume concentration of cracks and pores. Computational modeling results are presented. A time of solving the multi-physics problem using fine and coarse grids is shown.