A solenoidal basis is constructed to compute velocities using a certain finite element method for the Stokes problem. The method is conforming, with piecewise linear velocity and piecewise constant pressure on the Powell-Sabin split of a triangulation. Inhomogeneous Dirichlet conditions are supported by constructing an interpolating operator into the solenoidal velocity space. The solenoidal basis reduces the problem size and eliminates the pressure variable from the linear system for the velocity. A basis of the pressure space is also constructed that can be used to compute the pressure after the velocity, if it is desired to compute the pressure. All basis functions have local support and lead to sparse linear systems. The basis construction is confirmed through rigorous analysis. Velocity and pressure system matrices are both symmetric, positive definite, which can be exploited to solve their corresponding linear systems. Significant efficiency gains over the usual saddle-point formulation are demonstrated computationally.

翻译：本文构建了一种散度基函数，用于基于特定有限元方法求解Stokes问题中的速度场。该方法具有协调性，在三角剖分的Powell-Sabin分裂上采用分片线性速度场与分片常数压力场。通过构造插值算子作用于散度速度空间，支持非齐次Dirichlet边界条件。该散度基函数减少了问题规模，并从速度线性系统中消去了压力变量。同时构造了压力空间的一组基函数，可在速度求解后（如需计算压力）用于压力场计算。所有基函数均具有局部支撑性，可生成稀疏线性系统。基函数构造经过严格分析验证。速度与压力系统矩阵均具有对称正定性，这一特性可用于求解相应的线性系统。数值实验表明，该方法相较于传统鞍点问题格式具有显著的效率提升。