A Fourth-Order Kernel-Free Boundary Integral Method for Interface Problems (Communications in Computational Physics, 2023)
This paper presents a fourth-order Cartesian grid based boundary integralmethod (BIM) for heterogeneous interface problems in two and three dimensionalspace, where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions. The method reformulates thegoverning equation with interface conditions into boundary integral equations (BIEs)and reinterprets the involved integrals as solutions to some simple interface problemsin an extended regular region. Solution of the simple equivalent interface problems forintegral evaluation relies on a fourth-order finite difference method with an FFT-basedfast elliptic solver. The structure of the coefficient matrix is preserved even with theexistence of the interface. In the whole calculation process, analytical expressions ofGreen’s functions are never determined, formulated or computed. This is the noveltyof the proposed kernel-free boundary integral (KFBI) method. Numerical experimentsin both two and three dimensions are shown to demonstrate the algorithm efficiencyand solution accuracy even for problems with a large diffusion coefficient ratio.
Keywords: Elliptic interface problem; compact scheme; finite difference method; Cartesian grid method; kernel-free boundary integral method; boundary integral equation.