Immersed-Interface Finite-Element Methods for Elliptic and Elasticity Interface Problems

Abstract

The purpose of the research has been to develop a class of new finite-element methods, called immersed-interface finite-element methods, to solve elliptic and elasticity interface problems with homogeneous and non-homogeneous jump conditions. Simple non-body-fitted meshes are used. Single functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface. With such functions, the discontinuities across the interface in the solution and flux are removed; and equivalent elliptic and elasticity interface problems with homogeneous jump conditions are formulated. Special finite-element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions. Error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate. These methods are designed as an efficient component of the finite-element level-set methodology for fast simulation of interface dynamics that does not require re-meshing. Such simulation has been a powerful numerical approach in understanding material properties, biological processes, and many other important phenomena in science and engineering.