We consider radially symmetric capillary surfaces that are described by bounded generating curves. We use the arc-length representation of the differential equations for these surfaces to allow for vertical points and inflection points along the generating curve. These considerations admit capillary tubes, sessile drops, and fluids in annular tubes as well as other examples. We present a multi-scale pseudo-spectral method for approximating solutions of the associated boundary value problems based on interpolation by Chebyshev polynomials. The multi-scale approach is based on a domain decomposition with adaptive refinements within each sub-domain.

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