A numerical method for solving the elliptic and elasticity interface problems

Interface problems arise when dealing with physical problems composed of different materials or of the same material at different states. Because of the irregularity along interfaces, many common numerical methods do not work, or work poorly, for interface problems. Matrix-coefficient elliptic and elasticity equations with oscillatory solutions and sharp-edged interfaces are especially complicated and challenging for most existing methods. An accurate and efficient method is desired. In 1999, the boundary condition capturing method was proposed to deal with Poisson equations with interfaces whose variable coefficients and solutions may be discontinuous. In 2003, a weak formulation was derived. Built on previous work that solves elliptic interface problems with two domains in two dimensions, this dissertation improves the accuracy in the presence of sharp-edged interfaces and extends to elasticity interface problems with two domains in two dimensions, elliptic interface problems with three domains in two dimensions, and elliptic interface problems with two domains in three dimensions. The method used in this dissertation is a non-traditional finite element method. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear, satisfying the jump conditions across the interface. These two bases are different, which leads to the non-symmetric matrix generated by this method, but the resulting linear system of equations is shown to be positive definite under certain assumptions in all the four topics mentioned in this dissertation. This method has matrix coefficients and lower-order terms, and uses the non-body-fitting grid which makes it easy to deal with different kinds of interfaces, like the examples “Star”, “Happy face”, “Chess board”, to name a few. The methods used in this dissertation solve the non-smooth interface case and promise results for oscillatory solutions. Numerical experiments show that this method is second-order accurate in the L ∞ norm for piecewise smooth solutions

An interpolation matched interface and boundary method for elliptic interface problems

AbstractAn interpolation matched interface and boundary (IMIB) method with second-order accuracy is developed for elliptic interface problems on Cartesian grids, based on original MIB method proposed by Zhou et al. [Y. Zhou, G. Wei, On the fictious-domain and interpolation formulations of the matched interface and boundary method, J. Comput. Phys. 219 (2006) 228–246]. Explicit and symmetric finite difference formulas at irregular grid points are derived by virtue of the level set function. The difference scheme using IMIB method is shown to satisfy the discrete maximum principle for a certain class of problems. Rigorous error analyses are given for the IMIB method applied to one-dimensional (1D) problems with piecewise constant coefficients and two-dimensional (2D) problems with singular sources. Comparison functions are constructed to obtain a sharp error bound for 1D approximate solutions. Furthermore, we compare the ghost fluid method (GFM), immersed interface method (IIM), MIB and IMIB methods for 1D problems. Finally, numerical examples are provided to show the efficiency and robustness of the proposed method