On the uniform convergence of piecewise linear solutions to the equilibrium capillary surface equation

We consider piecewise linear solutions to the equilibrium capillary surface equations over a given triangulation of a multifaceted domain. It is shown that, under certain conditions, the gradients of these functions are bounded in refining the triangulation, i.e., when the maximum diameter of the triangles of the triangulation vanishes. This property holds if piecewise linear functions approximate the energy integral for a smooth function with required accuracy. As a consequence of the obtained properties, we get the uniform convergence of piecewise linear solutions to the exact solution of the equation of an equilibrium capillary surface with prescribed contact angle on the boundary.