Mathematical Sciences: High Order Accurate Numerical Methods for Interface Problems

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Osher 9626703 Many physical problems involve interfaces where two different materials contact with each other, or singular sources or dipoles are present along the interfaces immersed in the same fluid. Mathematically, interface problems usually lead to differential equations whose input data and solutions have discontinuities or non-smoothness across interfaces. Many numerical methods designed for smooth solutions do not work efficiently for interface problems. The investigator's colleague Zhilin Li combines the immersed interface method, a second order method for solving differential equations involving interfaces, with the level set approach, an efficient method for capturing moving fronts, to develop high order accurate and efficient numerical methods for interface problems. The project develops convergence and stability theory to provide theoretical justification for the proposed methods. Several specific interface problems, including elliptic and parabolic equations with fixed or moving interfaces, Hele-Shaw flow, and other applications, are studied in depth. Many important practical problems lead to differential equations in regions of 2- or 3-dimensional space that are geometrically complicated, and that contain interfaces across which the nature of the solution changes. These equations can rarely be solved exactly, and large-scale computation is required to obtain well-resolved solutions over multi-dimensional regions. The goal of this work is to develop efficient computational methods to approximate solutions of such problems. The approach is to combine two different methods with complementary strengths. To test the new method, it is applied to several problems representative of those with arising in practical applications.