We developed a kernel-free boundary integral method (KFBIM) for solving variable coefficients partial differential equations (PDEs) in a... Show moreWe developed a kernel-free boundary integral method (KFBIM) for solving variable coefficients partial differential equations (PDEs) in a doubly-connected domain. We focus our study on boundary value problems (BVP) and interface problems. A unique feature of the KFBIM is that the method does not require an analytical form of the Greenâ€™s function for designing quadratures, but rather computes boundary or volume integrals by solving an equivalent interface problem on Cartesian mesh. We decompose the problem defined in a doubly-connected domain into two separate interface problems. Then we evaluate integrals using a Krylov subspace iterative method in a finite difference framework. The method has second-order accuracy in space, and its complexity is linearly proportional to the number of mesh points. Numerical examples demonstrate that the method is robust for variable coefficients PDEs, even for cases when diffusion coefficients ratio is large and when two interfaces are close. We also develop two methods to compute moving interface problems whose coefficients in governing equations are spatial functions. Variable coefficients could be a non-homogeneous viscosity in Hele-Shaw problem or an uptake rate in tumor growth problems. We apply the KFBIM to compute velocity of the interface which allows more flexible boundary condition in a restricted domain instead of free space domain. A semi-implicit and an implicit methods were developed to evolve the interface. Both methods have few restrictions on the time step regardless of numerical stiffness. Theyalso could be extended to multi-phase problem, e.g., annulus domain. The methods have second-order accuracy in both space and time. Machine learning techniques have achieved magnificent success in the past decade. We couple the KFBIM 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. We also developed an efficient data generator for training and derived an empirical rule for data set size. In the future work, the model could be expanded to moving interface problems. The linear model will be replaced by neural network models, e.g., physics-informed neural networks (PINNs). Show less