We develop a spectral method for solving the incompressible generalized Navier--Stokes equations in the ball with no-flux and prescribed slip boundary conditions. The algorithm achieves an optimal complexity per time step of $\mathcal{O}(N\log^2(N))$, where $N$ is the number of spatial degrees of freedom. The method relies on the poloidal-toroidal decomposition of solenoidal vector fields, the double Fourier sphere method, the Fourier and ultraspherical spectral method, and the spherical harmonics transform to decouple the Navier--Stokes equations and achieve the desired complexity and spectral accuracy.

翻译：我们开发了一种光谱方法来解决球中无法压缩的通用导航-斯托克斯方程式问题,该方程式具有无流量和规定的滑行边界条件。算法每时步达到最高复杂性$mathcal{O}(N\log/2(N))$(美元),其中美元为自由空间度数。该方法依赖于单向矢量场、双向四向球法、Fourier和超球光谱法,以及球体调变异为除去导航-斯托克斯方程式,并达到预期的复杂度和光谱精度。