Structure preserving nodal continuous Finite Elements via Global Flux quadrature

TL;DR

The paper tackles the problem that standard stabilization techniques for hyperbolic acoustics on Cartesian grids fail to preserve discrete stationary states. It introduces Global Flux quadrature (GFq) within a tensor-product FE framework to construct constraint-compatible stabilizations (GFq-SUPG, GFq-OSS) that preserve the divergence-free equilibria and yield vorticity preservation, even achieving super-convergence on stationary states. The authors develop multi-dimensional GFq divergences, integrators, nodal projections, and a Deferred Correction time stepping scheme, and provide rigorous kernel and involution analyses alongside extensive numerical validation. The resulting approach improves long-time accuracy and reliability for hyperbolic PDE discretizations and holds promise for extension to non-homogeneous and nonlinear systems, including shallow water and Euler equations with gravity.

Abstract

Numerical methods for hyperbolic PDEs require stabilization. For linear acoustics, divergence-free vector fields should remain stationary, but classical Finite Difference methods add incompatible diffusion that dramatically restricts the set of discrete stationary states of the numerical method. Compatible diffusion should vanish on stationary states, e.g. should be a gradient of the divergence. Some Finite Element methods allow to naturally embed this grad-div structure, e.g. the SUPG method or OSS. We prove here that the particular discretization associated to them still fails to be constraint preserving. We then introduce a new framework on Cartesian grids based on surface (volume in 3D) integrated operators inspired by Global Flux quadrature and related to mimetic approaches. We are able to construct constraint-compatible stabilization operators (e.g. of SUPG-type) and show that the resulting methods are vorticity-preserving. We show that the Global Flux approach is even super-convergent on stationary states, we characterize the kernels of the discrete operators and we provide projections onto them.

Figures (14)

• Figure 1: Notation of the degrees of freedom for a function $q_h$ in element $E_{ij}$ for $\mathbb Q^4$ elements
• Figure 2: Oblique flow: convergence of $L^2$ error in $u$ w.r.t. the number of elements in $x$
• Figure 3: Simulation at time $T=100$ of the vortex \ref{['eq:definition_C6_vortex']} with $20\times 20$ cells and $\mathbb P^1$ elements for the SUPG (top) and SUPG--GF (bottom) schemes
• Figure 4: Simulation at time $T=100$ of the vortex \ref{['eq:definition_C6_vortex']} (only $p$) with $10\times 10$ cells and $\mathbb P^2$ elements for the SUPG (left) and SUPG--GF (right) schemes
• Figure 5: Norm of the discrete divergence of $\mathbf{v}$ for SUPG (${D}_x \otimes M_y u + M_x\otimes {D}_y v$) and SUPG-GFq (${D}_x \otimes (D_yI_y) u + (D_x I_x)\otimes {D}_y v$) as a function of time for different orders of accuracy

Theorems & Definitions (53)

• Definition 2.1: Finite differences
• Example 2.1
• Definition 2.2: High-order differences
• Example 2.2
• Definition 2.3: Composition in the high-order case
• Proposition 2.1
• Definition 2.4: Tensor-product high-order operators
• Proposition 2.2: Fourier transform of the tensor product in the high-order case
• proof
• Proposition 2.3