We present analysis of two lowest-order hybridizable discontinuous Galerkin methods for the Stokes problem, while making only minimal regularity assumptions on the exact solution. The methods under consideration have previously been shown to produce $H(\textrm{div})$-conforming and divergence-free approximate velocities. Using these properties, we derive a priori error estimates for the velocity that are independent of the pressure. These error estimates, which assume only $H^{1+s}$-regularity of the exact velocity fields for any $s \in [0, 1]$, are optimal in a discrete energy norm. Error estimates for the velocity and pressure in the $L^2$-norm are also derived in this minimal regularity setting. Our theoretical findings are supported by numerical computations.
翻译:我们对斯托克斯问题的两种最低顺序混合不连续的Galerkin方法进行了分析,同时只对确切的解决方案做出了最起码的规律性假设。审议中的方法以前曾显示产生H(\ textrm{div})$和零差异的近似速度。使用这些特性,我们得出不受压力影响的速度的先验误差估计数。这些误差估计数假定任何美元(0,1美元)的精确速度字段只有$+s}的规律性,在离散的能源规范中是最佳的。对于美元(L+2美元-norm)的速度和压力的误差估计数也是在这种最低限度的常规设置中得出的。我们的理论结论得到数字计算的支持。