不规则区域上三维 Stokes 和 Navier 方程的无核边界积分方法
Kernel Free Boundary Integral Method for 3D Stokes and Navier Equations on Irregular Domains
AI总结：
本文提出一种二阶精度的无核边界积分方法，通过基于笛卡尔网格的间接求值求解三维不规则区域上的 Stokes 和 Navier 边界值问题，消除了对 Green 函数解析式的依赖并避免了奇异性处理。
摘要：
本文提出了一种二阶精度的无核边界积分方法，用于求解三维不规则区域上的 Stokes 和 Navier 边界值问题。该方法在边界积分方程的框架下求解方程，其相应的离散形式良态，并通过 GMRES 方法求解。该方法的一个显著特征是，BIEs 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求值，该方法包括用 MAC 格式离散相应的简单界面问题，修正离散线性系统以减少界面附近较大的局部截断误差，以及通过 CG 方法结合基于 FFT 的 Poisson 求解器求解修正后的系统。处理奇异或超奇异边界积分不需要额外的工作或特殊求积，并且完全消除了对积分核 Green 函数解析表达式的依赖。给出了数值结果以证明基于笛卡尔网格方法的效率和精度。
英文摘要：
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.
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