A High-Order Kernel-Free Boundary Integral Method for the Biharmonic Equation on Irregular Domains

This work proposes second-order and fourth-order versions of a Cartesian grid based kernel-free boundary integral (KFBI) method for the biharmonic equation on both bounded irregular domains and singly periodic irregular domains. It is further development of the previous KFBI method for second-order elliptic PDEs. It reformulates boundary value problems of the fourth-order PDE as boundary integral equations of the first kind but the solution never needs to know the fundamental solution or Green's function of the elliptic operator. Evaluation of boundary or volume integrals in the solution of boundary integral equations is made by solving equivalent interface problems on Cartesian grids with standard finite difference methods and fast Fourier transform based solvers. The work decomposes the biharmonic equation into two Poisson equations. It assumes the solution to one Poisson equation, which has no boundary conditions, as the sum of a volume integral with a double layer boundary integral, and applies Green's third identity to derive a scalar boundary integral equation from the other Poisson equation that are subject to two boundary conditions. In the solution of the scalar boundary integral equation, each volume or boundary integral is evaluated with the KFBI method. Numerical examples are presented to demonstrate the solution accuracy and algorithm efficiency. A remarkable point of the work is that the nine-point compact difference scheme in dealing with each split second-order elliptic interface problem on irregular domains yields fourth-order accurate solution for the biharmonic equation.