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We report upon a theoretical study of collective magnonic modes in pairs of magnetic nano-elements with quasi-uniform magnetization. The mode spectrum and character are numerically computed for an individual isolated nano-element and then used to analytically calculate the splitting of the modes due to the inter-element magneto-dipole interaction. The results are compared with those obtained using direct simulations for the pairs of elements, yielding a generally good agreement. For the edge mode the interaction between the edges of the neighboring elements can exceed that between the edges of the same element, leading to softening of the mode profile and hence to the violation of the assumptions of the analytical approach. The softening has to be taken into account in the interpretation of dynamical studies of closely packed arrays of magnetic elements (magnonic crystals). V

The dispersion relations of collective oscillations of the magnetic moment of magnetic dots arranged in square-planar arrays and having magnetic moments perpendicular to the array plane are calculated. The presence of the external magnetic field perpendicular to the plane of array, as well as the uniaxial anisotropy for single dot are taken into account. The ferromagnetic state with all the magnetic moments parallel, and chessboard antiferromagnetic state are considered. The dispersion relation yields information about the stability of different states of the array. There is a critical magnetic field below which the ferromagnetic state is unstable. The antiferromagnetic state is stable for small enough magnetic fields. The dispersion relation is non-analytic as the value of the wave vector approaches zero. Non-trivial Van Hove anomalies are also found for both ferromagnetic and antiferromagnetic states.