  We consider prescribed mean curvature equations whose solutions are minimal
surfaces, constant mean curvature surfaces, or capillary surfaces. We consider
both Dirichlet boundary conditions for Plateau problems and nonlinear Neumann
boundary conditions for capillary problems and we consider domains in
$\mathbf{R}^2$ to be rectangles, disks, or annuli.
  We present spectral methods for approximating solutions of the associated
boundary value problems. These are either based on Chebyshev or
Chebyshev-Fourier methods depending on the geometry of the domain. The
non-linearity in the prescribed mean curvature equations is treated with a
Newton method. The algorithms are designed to be adaptive; if the prescribed
tolerances are not met then the resolution of the solution is increased until
the tolerances are achieved. 22
