Structure preserving nodal continuous Finite Elements via Global Flux quadrature

TL;DR

This paper introduces a novel Global Flux quadrature-based framework for finite element methods that preserves stationary states and vorticity in hyperbolic PDEs, improving stability and accuracy on Cartesian grids.

Contribution

It develops a new surface-integrated operator framework that enables the construction of constraint-preserving stabilization methods for hyperbolic PDEs.

Findings

Global Flux approach is super-convergent on stationary states

Constructed constraint-compatible stabilization operators

Methods preserve vorticity and stationary states

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.