A kernel-free boundary integral method (KFBIM) for solving elliptic PDEs

Abstract: In this talk, we focus our study on elliptic boundary value problems (BVP) with variable coefficients. Unlike the traditional integral methods, a unique feature of the KFBIM is that the method does not require an analytical form of the Green’s function for designing quadratures, but rather computes boundary or volume integrals by solving an equivalent interface problem on Cartesian mesh. We first introduce ideas and procedures behind the KFBIM method and then couple it with supervised learning algorithms to improve efficiency. In the KFBIM, we apply a finite difference scheme to find dipole density of the boundary integral iteratively, which is quite costly. We train a linear model to replace the finite difference solver in GMRES iterations. The cost, measured in CPU time, is significantly reduced.