Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows

TL;DR

This work addresses the robust numerical discretization of the time-dependent incompressible Navier–Stokes equations by reinterpreting the momentum as a differential 1-form $\omega$ and applying a mesh-based semi-Lagrangian discretization within the finite-element exterior calculus (FEEC) framework. It introduces first- and second-order semi-Lagrangian schemes (with energy-conserving variants) that approximate the material derivative via backward-flow pullbacks and a projection operator $\mathcal{I}_{h,p}$, achieving stability as viscosity $\varepsilon\to0$ and enabling inviscid Euler simulations. The authors establish second-order convergence in space and time, enforce energy conservation through a Lagrange multiplier in the conservative variant, and demonstrate multiple numerical experiments (Taylor–Green vortices, rotating hump, 3D transient, lid-driven cavity, and complex domains) that illustrate accurate energy behavior and convergence. The approach provides a structure-preserving alternative to standard Eulerian methods by integrating boundary conditions and incompressibility into discrete 1-form spaces, with potential impact on high-Reynolds-number and free-boundary flow simulations.

Abstract

We develop a mesh-based semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions recast as a non-linear transport problem for a momentum 1-form. A linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible Euler flows. Conservation of energy and helicity are enforced separately.

Figures (16)

• Figure 1: Illustration of small edges (a) and corresponding local shape functions (b) for the unit triangle.
• Figure 2: For $p=2$ and $d=2$ the matrix $\mathcal{M}$ corresponding to the 2-simplex $K$ in (a) has the form given in (b). Each row and column in $\mathcal{M}$ is associated to a small edge in (a). Each submatrix in (b) describes the interactions between edges with the same color in (a). The gray submatrix is an exception as it describes the one-directional interaction between the small edges that lie on a big edge and the small edges that lie in the interior. For $d=3$, $\mathcal{M}$ has the structure as shown in \ref{['fig:matrixBlockStructure3D']}, where the purple submatrices are $2\times 2$ invertible matrices and the orange submatrices are $3\times 3$ matrices of rank 2.
• Figure 3: Edge $e$ (in red) is transported using the flow $\boldsymbol{\beta}$ (in blue). The exact transported edge $X_{\tau}(e)$ and the approximate transported edge $\bar{X}_{\tau}(e)$ are given in orange and green.
• Figure 4: The red line indicates the line that spans multiple elements. On the left we see the reference triangle associated with the yellow element in the mesh on the right.
• Figure 5: A coarse triangulation of $\Omega = \{\boldsymbol{x}\in\mathbb{R}^2; ||x||<1\}$ with the velocity field $\boldsymbol{u}=[-y,x]^T$ satisfying $\boldsymbol{u}\cdot\boldsymbol{n}=0$. Despite the vanishing normal components of the velocity, the blue edge gets transported out of the domain to the green edge.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3