Elliptic interface problems appear in many science and engineering problems. In this thesis, we mainly focus on numerical approaches for two of such problems. One relates to an elliptic operator defined on surfaces, while the other one is to solve a piecewisely defined elliptic equation with certain jump condition imposed on the boundary between different regions.

We firstly focus on solving elliptic equations on surfaces that are represented by point clouds. Motivated by solving time-dependent diffusion equations on surfaces, we develop a new discretization of the Laplace-Beltrami (LB) operator on point cloud. This new discretization assigns more weight to the diagonal elements of the discretized matrix so that the matrix is better conditioned. We then combine the new discretization of t...[ Read more ]

Elliptic interface problems appear in many science and engineering problems. In this thesis, we mainly focus on numerical approaches for two of such problems. One relates to an elliptic operator defined on surfaces, while the other one is to solve a piecewisely defined elliptic equation with certain jump condition imposed on the boundary between different regions.

We firstly focus on solving elliptic equations on surfaces that are represented by point clouds. Motivated by solving time-dependent diffusion equations on surfaces, we develop a new discretization of the Laplace-Beltrami (LB) operator on point cloud. This new discretization assigns more weight to the diagonal elements of the discretized matrix so that the matrix is better conditioned. We then combine the new discretization of the LB operator with the Fast Sweeping Method to solve eikonal equations on surfaces represented by the Grid Based Particle Method (GBPM). We also develop an efficient method to extract the geodesic distance once we have the viscosity solution of the eikonal equation. Finally, we incorporate a recently developed elliptic Finite Element (FEM) solver to take care of the jump condition in the second elliptic interface problem where the interface is represented by GBPM and the Closest Point Method (CPM). The proposed framework and algorithms can solve differential equation on dynamic interfaces and be used in various important applications including multiphase flow modelings in the future.