Wenjun Ying: Recent developments of a potential theory based Cartesian grid method for elliptic PDEs on irregular domains

Abstract: This talk will be on a potential theory based Cartesian grid method for elliptic PDEs on irregular domains. The method solves a boundary value or interface problem of PDE in the framework of second-kind Fredholm boundary integral equations. It avoids some limitations of the traditional boundary integral method. It does not need to know or compute the fundamental solution or Green's function of the PDE. Instead, it allows the solution of variable coefficients and nonlinear PDEs. The method evaluates boundary and volume integrals involved indirectly by solving equivalent but much simpler interface problems on Cartesian grids, based on properties of single, double layer boundary integrals and volume integrals in potential theory. In addition to its taking advantage of the well-conditioning property of the second-kind Fredholm boundary integral equations, the method makes full use of fast solvers on Cartesian grids. The Cartesian grid method can also accurately compute nearly singular and hypersingular boundary integrals. In this talk, I will present recent developments of the method.