A Fast Cartesian Grid-Based Integral Equation Method for Unbounded Interface Problems with Non-Homogeneous Source Terms (Communications in Computational Physics, 2023)
This work presents a fast Cartesian grid-based integral equation method forunbounded interface problems with non-homogeneous source terms. The unboundedinterface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally. This work overcomes two difficulties.The first difficulty is the evaluation of singular integrals. Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problemson rectangular domains, which are solved with FFT-based finite difference solvers. Thesecond one is the expensive computational cost for volume integrals. Despite the useof efficient interface problem solvers, the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem.The problem is alleviated by introducing an auxiliary circle as a bridge to indirectlyevaluate boundary conditions. Since solving boundary integral equations on a circular boundary is so accurate, one only needs to select a fixed number of points for thediscretization of the circle to reduce the computational cost. Numerical examples arepresented to demonstrate the efficiency and the second-order accuracy of the proposednumerical method.
Keywords: Interface problem; unbounded domain; boundary integral equation; kernel-free method; auxiliary circle; Cartesian grid method; fast algorithm.