On the uniform convergence of piecewise linear solutions of an equilibrium capillary surface equation
Aleksey A. Klyachin · Journal of Applied and Industrial Mathematics · 2015
We consider piecewise-linear solutions of the equilibrium equation of a capillary surface over a given triangulation of a multifaceted (polygonal) region. It is shown that, under certain conditions, the gradients of these functions remain bounded in absolute value when the partition is refined; i.e., the maximum diameter of the triangulation triangles tends to zero. This property is fulfilled if the piecewise-linear functions approximate the energy integral of a smooth function with the required accuracy. Some consequence of the obtained properties is the uniform convergence of the piecewise-linear solutions to the exact solution of the equilibrium capillary surface equation with the boundary condition in the form of a prescribed contact angle.