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：本文提出了一种时空连续伽辽金有限元方法，用于求解不可压缩Navier-Stokes方程。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。我们对稳定化格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力学中的两个基准问题：顶盖驱动方腔流动和圆柱绕流。通过现有文献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极大尺度的时空块。