Kernel-free Boundary Integral Method For Biharmonic Boundary Value Problems

The aim of this paper is to extend the kernel-free boundary integral (KFBI) meth-ods by Ying to solve the biharmonic boundary value problems. We decompose thebiharmonic BVPs into two elliptic BVPs, and form one boundary integral equationto solve the problem. The boundary integral equations reformulated from the bihar-monic BVPs are solved iteratively with the GMRES method. The KFBI methods canbe used to solve the elliptic boundary value problems, then can be used to solve thebiharmonic BVPs. Due to the advantages of KFBI methods, we can avoid the need toknow the analytical expressions of Green’s functions, can solve the biharmonic BVPson complex domains, can generate a structured grids by embedding in a larger regu-lar domain, can use the fast Fourier transform (FFT) based Poisson/Helmholtz solverson the structured grids. Numerical results demonstrating the efciency and accuracyof the KFBI methods for solving biharmonic BVPs are presented. The KFBI methodshows a second-order convergence rate in accuracy for all of the tested biharmonicBVPs. Finally, we also tested the two-dimensional incompressible steady Stokes fowproblem, the problem reformulated into a biharmonic problem on a single connectedregion can be well solved.