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

TL;DR

This paper develops a finite element framework that preserves the eigenvalue range of symmetric tensor fields in time-dependent convection–diffusion problems. By formulating a variational inequality on a convex bound set $\\mathbb{S}_d^{\\epsilon,\\kappa}$ and enforcing nodewise eigenvalue clipping via a projection, the method integrates with a baseline continuous interior penalty discretisation and implicit Euler time stepping to achieve unconditional stability and optimal-order convergence. A rigorous stability and error analysis is provided, supported by numerical experiments showing effective bound preservation in convection-dominated regimes and with non-smooth data. The approach directly enforces physical admissibility at the degrees of freedom, offering robustness and a linear structure in the unconstrained limit, with promising extensions to higher-order time schemes and nonlinear problems.

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.

Figures (8)

• Figure 1: Domain $\Omega$ with velocity field ${\boldsymbol{\beta}} = (-y, x)^{T}$ and inflow boundary $\Gamma_{\text{in}}$.
• Figure 2: Left: Convergence with respect to the time step size $\Delta t$ for $\mathbb{P}_{1}$ elements with $h=1/50$. Right: Convergence with respect to the mesh size $h$ for the BP--Euler method using $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ elements with $\Delta t = 1/500$.
• Figure 3: Solution of \ref{['example22']} at $T=4$ with discontinuous inflow data: minimal and maximal eigenvalues obtained with the CIP--Euler and BP--Euler schemes, together with cross-sections taken along $y=x$ ($\mathbb{P}_{1}$ elements).
• Figure 4: Solution of \ref{['example22']} at $T=4$ with discontinuous inflow data: minimal and maximal eigenvalues obtained with the CIP--CN and BP--CN schemes, together with cross-sections taken along $y=x$ ($\mathbb{P}_{1}$ elements).
• Figure 5: Solution of \ref{['example22']} at $T=4$ with discontinuous inflow data: minimal and maximal eigenvalues obtained with the CIP--Euler and BP--Euler schemes, together with cross-sections taken along $y=x$ ($\mathbb{P}_{2}$ elements).

Theorems & Definitions (12)

• Remark 3.1: Componentwise interpretation of the transport terms
• Remark 3.2: Tensor-valued basis construction
• Remark 3.3: Convexity of the admissible set
• Theorem 4.1: Well-posedness of the fully discrete variational inequality
• proof
• Lemma 5.1: Discrete Grönwall lemma
• Lemma 5.2: Energy stability
• proof
• Theorem 5.3: A priori error estimate
• proof