A second-order accurate kernel-free boundary integral method is presented for Stokes and Navier boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green's functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method.

翻译：本文提出了一种针对三维不规则区域上Stokes和Navier边值问题的二阶精确无核边界积分方法。该方法在边界积分方程的框架下求解方程，其对应的离散形式具有良好的条件数，并通过GMRES方法求解。该方法的一个显著特征是，边界积分方程中遇到的边界或体积积分通过基于笛卡尔网格的方法间接计算，包括：采用MAC格式离散相应的简单界面问题，修正离散线性系统以减小界面附近较大的局部截断误差，通过共轭梯度法与基于快速傅里叶变换的泊松求解器相结合求解修正后系统。无需额外工作或特殊求积方法处理奇异或超奇异边界积分，且完全消除了对格林函数积分核解析表达式的依赖。数值结果验证了基于笛卡尔网格方法的效率和精度。