AI 中文总结
研究具有非光滑数据的纳维-斯托克斯方程，引入正则化公式，利用泰勒-胡德\(P2P1\)元构建后验误差估计器，导出误差上下界，为低正则性狄利克雷边界数据的不可压缩流自适应有限元方法提供严格基础。
AI 中文摘要
我们研究具有狄利克雷边界数据且处于\(L^2\)空间的定常纳维-斯托克斯方程，此情形下解的正则性有限，阻碍了标准后验误差估计技术的直接应用。为解决该问题，我们引入正则化公式，得到原问题的适定近似并允许协调有限元离散化。利用泰勒-胡德\(P2P1\)元，构建基于残差的后验误差估计器，并在数据的适当小性假设下建立其可靠性和有效性。我们在合适范数下导出可计算的上下界，将估计器与原纳维-斯托克斯问题精确解及其有限元近似之间的误差相关联，表明估计器准确反映有限元误差。这些结果为具有低正则性狄利克雷边界数据的不可压缩流的自适应有限元方法的分析和实现提供了严格基础。
英文摘要
We study the stationary Navier-Stokes equations with Dirichlet boundary data in L2, a setting in which the limited regularity of the solution prevents the direct application of standard a posteriori error estimation techniques. To address this issue, we introduce a regularized formulation that yields a well-posed approximation of the original problem and admits a conforming finite element discretization. Using Taylor-Hood P2P1 elements, we construct a residual-based a posteriori error estimator and establish its reliability and efficiency under suitable smallness assumptions on the data. We derive computable upper and lower bounds in an appropriate norm that relate the estimator to the error between the exact solution of the original Navier-Stokes problem and its finite element approximation, showing that the estimator accurately reflects the finite element error. These results provide a rigorous foundation for the analysis and implementation of adaptive finite element methods for incompressible flows with low-regularity Dirichlet boundary data.
