An efficient numerical self-consistent field theory (SCFT) algorithm is developed for polymer systems subject to nonperiodic boundary conditions. The method involves collocation with a Chebyshev polynomial basis in order to solve the modified diffusion equations of SCFT and relax the fields experienced by polymer segments to a consistent mean-field solution. For Dirichlet boundary conditions on a block copolymer film, we demonstrate that the Chebyshev scheme preserves spectral accuracy, unlike existing methods based on collocation with a Fourier sine basis or finite differences. This advantage is shown to translate into significant benefits for accuracy, computational cost, and memory requirements in high-resolution SCFT simulations of meso and nanostructured polymer thin films. As a nontrivial application of the new Chebyshev method, the static wetting behavior of a two-dimensional fluid polymer droplet on a solid surface was investigated via SCFT simulations. The interaction between the surface and polymer segments is modeled with a local contact Flory-Huggins parameter that can be mapped to a surface tension. Contact angles obtained from simulations with varying surface and interfacial tensions are shown to be consistent with Young's equation. The results demonstrate that Chebyshev-based SCFT is a viable and efficient technique for exploring complex morphologies and equilibrium phenomena in polymer thin films.