In the present work, the Fast Multipole Boundary Element Method (FMM / BEM) for solving three-dimensional incompressible fluid flow problems governed by the Navier–Stokes equations is proposed. With the velocity-vorticity formulation the pressure gradient is eliminated from the equations. The kinematics equation, related to the velocity field satisfies continuity and provides a direct boundary condition for the vorticity equation. The single-domain approach is used for the discretization of the entire computational volume. The system of equations is compressed into two vectors and a preconditioner matrix, which is negligible in size. The involved unknowns are velocities, vorticities, and boundary vorticity fluxes. The two governing equations are coupled together in a convergent Newton–Raphson iteration scheme, successfully used for the solution of 3D fluid flow problems on a 32 GB memory computer. The degrees of freedom of the benchmark problems are above to 300000, which is an unreachable limit for the conventional single-domain BEM.

在目前的工作中，提出了用于解决由Navier–Stokes方程控制的三维不可压缩流体流动问题的快速多极边界元方法（FMM / BEM）。采用速度涡度公式，可以从方程中消除压力梯度。与速度场有关的运动学方程满足连续性，并为涡度方程提供了直接的边界条件。单域方法用于整个计算量的离散化。方程组被压缩为两个向量和一个前置条件矩阵，其大小可以忽略。涉及的未知数是速度，涡度和边界涡度通量。这两个控制方程在牛顿-拉夫森收敛方案中耦合在一起，在32 GB内存计算机上成功用于解决3D流体流动问题。基准问题的自由度超过300000，这对于常规的单域BEM来说是一个无法达到的限制。