Discourse On The Matched Interface And Boundary Method (MIB)

The numerical solution of elliptic equation with discontinuous coefficients (also known as:elliptical interface problem) is of significant importance in applied mathematics and scientific computation, due to their widespread applications in fluid dynamics, electro-magnetics, material science, and biological systems. Peskin has pioneered the research of these problems. The immersed boundary method (BBM), proposed by him, is used to study the cardiac fluid dynamic. Since then a series of important methods has been proposed by researchers, such as:integral equation approach, immersed interface method, ghost fluid method, finite-volume-based methods, and so on. The matched interface and boundary method (MEB) is the first computational method that is of second order accuracy for elliptic interface problem with arbitrarily complex interface and singular sources. In molecular mathematical biology, the MIB method is proved to be a powerful tool. And it contributes a lot to areas like, potential driven molecular surface, electrostatic potential on protein surface, ion channel behaviors and so on. In the dissertation, the basic ideas of the MIB method are discussed. Further, the application of MIB method in multi-material interface problem, the adaptively deformed mesh method, and Galerkin formulation is demonstrated in great detail.In multi-material elliptic interface problem, the geometric singularities originated from the interfaces joining together are amplified as there are three or more subdomains intersecting with each other. And several kinds of jump conditions can coexist in the same point. We thus invent various schemes to take care of different geometries. The accuracy and stability of these schemes are further verified with designed numerical tests.The combination of MIB method and adaptively deformed mesh method is also discussed. The basic idea of the adaptively deformed mesh method is to concentrate mesh nodes in special areas, like where the function values change dramatically, to improve the accuracy. Due to the existence of the interface, we need to incorporate interface jump conditions into mesh deformed functions. Based on the interface geometry or the variation of the function values, two schemes are developed and then validated by rigorous numerical tests.The MIB Galerkin representation is a related important problem. To ensure continuity of the base functions cross the interface, overlapping elements called MIB elements, are defined. In the local area of the fictitious point, we construct a pair of second order polynomials. Their coefficients can be represented by function values and jump conditions. The schemes of fictitious values are then determined by these coefficients. We employ the traditional MIB technique that is to replace of the extended value with fictitious value. And the final linear matrix equation is delivered.Finally, we summarize the application of the MIB method and the whole thesis.Most of the research presented here in this thesis has been completed in the Department of Mathematics, Michigan State University, under the guidance of Prof.Guowei Wei.The research material presented in this thesis is mainly adopted from related publications.