For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.

中文翻译：

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斯托克斯方程有限元解的显式后验和先验误差估计

对于 2D 和 3D 域上的斯托克斯方程，为有限元解新开发了显式后验和先验误差估计。利用扩展超圆法和 Scott-Vogelius 有限元方案解决了处理 Stokes 方程无发散条件的困难。由于误差估计中的所有项都有明确的值，通过进一步应用区间算法和经过验证的计算算法，计算结果为近似误差提供了严格的估计。作为所提出的误差估计的应用，考虑了斯托克斯算子的特征值问题并获得了特征值的严格界限。通过在凸和非凸 3D 域上求解斯托克斯方程，证明了所提出的误差估计的效率。