Numerical study of unbounded capillary surfaces

Yasunori Aoki and Hans De Sterck

Vol. 267 (2014), No. 1, 1–34

DOI: 10.2140/pjm.2014.267.1

Abstract

Unbounded capillary surfaces in domains with a sharp corner or a cusp are studied. It is shown how numerical study using a proposed computational methodology leads to two new conjectures for open problems on the asymptotic behavior of capillary surfaces in domains with a cusp. The numerical methodology contains two simple but important ingredients, a change of variable and a change of coordinates, which are inspired by known asymptotic approximations for unbounded capillary surfaces. These ingredients are combined with the finite volume element or Galerkin finite element methods. Extensive numerical tests show that the proposed computational methodology leads to a global approximation method for singular solutions of the Laplace–Young equation that recovers the proper asymptotic behavior at the singular point, is more accurate and has better convergence properties than numerical methods considered for singular capillary surfaces before. Using this computational methodology, two open problems on the asymptotic behavior of capillary surfaces in domains with a cusp are studied numerically, leading to two conjectures that may guide future analytical work on these open problems.

Keywords

singularity, asymptotic analysis, nonlinear elliptic PDE, Laplace–Young equation, finite element method

Mathematical Subject Classification 2010

Primary: 35J25, 65N30, 76B45, 35J75

Milestones

Received: 13 June 2012

Revised: 18 September 2012

Accepted: 24 September 2012

Published: 22 December 2013