The problem of a quantum-mechanical description of a near-barrier fusion of heavy nuclei, that occurs at strong coupling of their relative motion to surface vibrations, is analyzed. To this end, an efficient finite-element method is proposed for numerically solving coupled Schrödinger equations with boundary conditions corresponding to total absorption. The method allows us to eliminate the instabilities in the numerical solutions that appear at a large number of coupled channels in some reactions. To illustrate the validity of our approach, the results of fusion cross section of the ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$ and ${}^{36}\mathrm{S}+{}^{48}\mathrm{Ca}$ reactions have been re-examined. The obtained results demonstrate a remarkable agreement with the available experimental data. It is found that experimental data can be reproduced with the use of the Woods-Saxon potential, without introducing the repulsive cores. It appears that the fusion cross sections at deep sub-barrier energies are sensitive to the potential pocket profile.

We analyze the origin of the unexpected deep subbarrier heavy-ion fusion hindrance in ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$, ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$, and ${}^{28}\mathrm{Si}+{}^{64}\mathrm{Ni}$ reactions. Our analysis is based on the improved coupled-channels description, implemented by means of the finite element method. With the aid of the Woods-Saxon potential the experimental cross sections and the $S$ factors of these reactions are remarkably well reproduced within the sudden approximation approach. We found that accounting for the nondiagonal matrix elements of the coupling matrix, traditionally neglected in the conventional coupled-channels approaches in setting the left boundary conditions inside the potential pocket, and its minimal value are crucially important for the interpretation of experimental data. We found as well a good agreement with the general trend of the experimental data for the $S$ factor of the fusion reaction ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$, which has no pronounced maximum for this system.