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As scattering-scanning near-field optical microscopy (s-SNOM) continues to grow in prominence, there has been great interest in modeling the near-field light-matter interaction to better predict experimental results. Both analytical and numerical models have been developed to describe the near-field response, but thus far models have not incorporated the full range of phenomena accessible. Here, we present a finite element model (FEM), capable of incorporating the complex physical and spatial phenomena that s-SNOM has proved able to probe. First, we use electromagnetic FEM to simulate the multipolar response of the tip and illustrate the impact of strong coupling on signal demodulation. We then leverage the multiphysics advantage of FEM to study the electrostatic effect of metallic tips on semiconductors, finding that THz s-SNOM studies are most impacted by this tip-induced band-bending. Our model is computationally inexpensive and can be tailored to specific nanostructured systems and geometries of interest.