Nonlocal optical response of materials, important at the nanometric scale, influences numerous optical phenomena, such as electromagnetic field confinement and spectral characteristics of plasmonic resonances. Here, we present a general phenomenological approach to account for nonlocal material polarizabilities in nanoscale metal particles. The problem of nonlocal plasmonic resonances is formulated by an integro-differential equation in a space domain and solved by adopting its weak form, implemented in the finite element method, thus, dispensing with the requirements on additional boundary conditions. As an example, nonlocal smearing effects in plasmonic nanorods of various cross sections and nanotubes have been considered. Clear signature of nonlocality manifests itself in the interference fringes in the potential profile and a significant frequency shift of the localized surface plasmon resonances. These effects are especially important for nanoparticles with geometrical features comparable to the de Broglie wavelengths of electrons participating in the light-matter interactions. The proposed method provides a universal tool for phenomenological account of nonlocalities of any kind with the only requirement of linearity in system's response.