Speaker
Yasunori Aoki, Applied Mathematics, University of Waterloo
Abstract:
In everyday life, it is often safe to assume that the surface of water is almost flat; however, careful observation can tell us that the surface of water in a container can exhibit complicated geometry near the interface where the water meets the container. One of the most extreme examples of complicated geometry can arise when the container has a sharp corner. In that case, the geometry of the liquid surface can appear as an unbounded singularity of the solution of the modeling partial differential equation, the Laplace-Young equation. The singularity of the solution of this PDE is well studied and the asymptotic series approximation of the solution is known. However, the asymptotic series approximation always comes with a fine print warning “the approximation is only valid in a sufficiently small neighbourhood of the singularity”, hence it is only a local approximation. We wish to obtain a global approximation of the solution through finite element approximation; however, it is also known that the singularity of the solution spoils the accuracy of a standard finite element approximation and the approximation cannot reproduce the singularity accurately. In this talk, we propose a numerical methodology to approximate these singular solutions that utilizes a-priori knowledge about the asymptotic growth order of the solution. We show that an accurate global approximation can be obtained through this numerical approach and verify that the numerical approximation has the correct asymptotic behaviour near the singularity.