We present a space-time continuous-Galerkin finite element method for solving incompressible Navier-Stokes equations. To ensure stability of the discrete variational problem, we apply ideas from the variational multi-scale method. The finite element problem is posed on the ``full" space-time domain, considering time as another dimension. We provide a rigorous analysis of the stability and convergence of the stabilized formulation. And finally, we apply this method on two benchmark problems in computational fluid dynamics, namely, lid-driven cavity flow and flow past a circular cylinder. We validate the current method with existing results from literature and show that very large space-time blocks can be solved using our approach.

翻译：本文提出了一种用于求解不可压缩Navier-Stokes方程的时空连续Galerkin有限元方法。为确保离散变分问题的稳定性，我们应用了变分多尺度方法的思想。该有限元问题建立在“完整”的时空域上，将时间视为另一个维度。我们对稳定化公式的稳定性和收敛性进行了严格分析。最后，我们将该方法应用于计算流体动力学中的两个基准问题：顶盖驱动方腔流和圆柱绕流。我们利用文献中的现有结果验证了当前方法的有效性，并表明我们的方法能够求解非常大的时空块。