Solving fluid flow problems in space-time with multiscale stabilization: formulation and examples

Abstract

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.

Figures (30)

• Figure 1: Schematic depiction of the space-time domain $U = \Omega \times I_T$, where $\Omega \subset \mathbb{R}^{d}$, and $I_T = [0,T] \subset \mathbb{R}^{+}.$
• Figure 2: $\Omega \subset \mathbb{R}^2,\ {U} \subset \mathbb{R}^3$
• Figure 3: $\Omega \subset \mathbb{R}^3,\ {U} \subset \mathbb{R}^4$
• Figure 5: (Left) A cubic space-time mesh in $(2D+t)$. Spatial coordinates are $(x,y)$, and $z$ denotes time. (Right) domain decomposition in space-time using 8 sub-domains.
• Figure 6: $\left\| u_x^h-u_x \right\|_{L^{2}(U)}$

Theorems & Definitions (20)

• Remark 2.1
• Remark 2.2
• Remark 2.3: Integration by parts
• Remark 2.4
• Remark 2.5
• Remark 2.6: Differences with respect to method of lines discretizations
• Remark 2.7
• Remark 2.8
• Lemma 3.1: Coercivity
• proof