In this talk, we will introduce an augmented matched interface and boundary (AMIB) method for solving elliptic interface and boundary value problems. The AMIB method provides a uniform fictitious domain approach to correct the fourth order central difference near interfaces and boundaries. In treating a smoothly curved interface, zeroth and first order jump conditions are enforced repeatedly to generate fictitious values surrounding the interface. Different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations, can be imposed to generate fictitious values outside boundaries. Based on fictitious values at interfaces and boundaries, the AMIB method reconstructs Cartesian derivative jumps as additional unknowns and forms an enlarged linear system. In the Schur complement solution of such system, the FFT algorithm will not sense the solution discontinuities, so that the discrete Laplacian can be efficiently inverted. In our 2D tests, the FFT-based AMIB not only achieves a fourth order convergence in dealing with interfaces and boundaries, but also produces an overall complexity of O(n^2 log n) for a n-by-n uniform grid. Moreover, the AMIB method can provide fourth order accurate approximations to solution gradients and fluxes.

Dr. Shan Zhao received the B.S. degree in mathematics from Lanzhou University, China, in 1997, and the Ph.D. degree in scientific computing from National University of Singapore, in 2003. From 2003 to 2006, he was a postdoctoral fellow with the Michigan State University. Shan has been on the faculty of University of Alabama since 2006, and was promoted to the rank of full professor in 2015. His research interest includes the development of high-order and fast algorithms for PDE interface problems and math modeling in molecular biophysics. He has published more than 70 papers, which have been cited more than 2000 times according to the Google Scholar. Since 2006, his research activities have received frequent funding from the National Science of Foundation (NSF) with a total of six grants. For more information, one can refer to http://szhao.people.ua.edu/.