A finite element method preserving the eigenvalue range of symmetric tensor fields

TL;DR

This paper introduces a finite element method that ensures eigenvalue bounds of tensor solutions are preserved during simulations of convection-diffusion equations, improving stability and accuracy in tensor-valued PDEs.

Contribution

It formulates a variational inequality approach with eigenvalue constraints at degrees of freedom, incorporating a projection-based eigenvalue truncation within a finite element framework.

Findings

The method maintains eigenvalue bounds at all degrees of freedom.

It achieves unconditional stability with implicit Euler discretization.

Numerical results confirm accurate solutions in convection-dominated regimes.

Abstract

This paper presents a finite element method that preserves (at the degrees of freedom) the eigenvalue range of the solution of tensor-valued time-dependent convection--diffusion equations. Starting from a high-order spatial baseline discretisation (in this case, the CIP stabilised finite element method), our approach formulates the fully discrete problem as a variational inequality posed on a closed convex set of tensor-valued functions that respect the same eigenvalue bounds at their degrees of freedom. The numerical realisation of the scheme relies on the definition of a projection that, at each node, performs the diagonalisation of the tensor and then truncates the eigenvalues to lie within the prescribed bounds. The temporal discretisation is carried out using the implicit Euler method, and unconditional stability and optimal-order error estimates are proven for this choice. Numerical experiments confirm the theoretical findings and illustrate the method's ability to maintain eigenvalue constraints while accurately approximating solutions in the convection-dominated regime.