A nodally bound-preserving finite element method for hyperbolic convection-reaction problems

TL;DR

This work develops a nodally bound-preserving finite element method for hyperbolic convection-reaction problems by recasting the discrete system as a variational inequality on a nodal-bound convex set $K_h$, ensuring pointwise bounds at nodes. The method employs SUPG stabilization via the bilinear form $a_h$ and proves an optimal best-approximation error bound in a natural NP-stability norm, with convergence rate $h^{\min\{k+1,r\}-\tfrac{1}{2}}$ for $u\in H^{r}(\Omega)$. The authors extend the framework to nonlinear reactions $|u|^{p-2}u$ ($1<p\le2$) using monotone-operator theory and a quasi-norm $\|\cdot\|_{(w,p)}$, obtaining analogous error control. They implement projection-based and reduced-space active-set solvers within Firedrake-PETSc to solve the variational inequalities and demonstrate through five numerical examples that the bound-preserving approach maintains physical bounds, suppresses nonphysical oscillations, and achieves the predicted convergence behavior, even under discontinuous data or nonlinearities. The work lays groundwork for applying bound-preserving FE methods to kinetic and non-Newtonian problems, with emphasis on stability, accuracy, and scalability in practical simulations.

Abstract

In this article, we present a numerical approach to ensure the preservation of physical bounds on the solutions to linear and nonlinear hyperbolic convection-reaction problems at the discrete level. We provide a rigorous framework for error analysis, formulating the discrete problem as a variational inequality and demonstrate optimal convergence rates in a natural norm. We summarise extensive numerical experiments validating the effectiveness of the proposed methods in preserving physical bounds and preventing unphysical oscillations, even in challenging scenarios involving highly nonlinear reaction terms.

Figures (7)

• Figure 1: Approximation errors for the bound-preserving finite element method in the full SUPG norm ${\left\vert\left\vert\left\vert \cdot \right\vert\right\vert\right\vert}$. Polynomial degrees $k =1,2$ shown, yielding expected convergence rates of $3\slash 2$ and $5 \slash 2$ respectively in the smooth case. As expected when the solution has minimal regularity, theoretical rates are not attained and there is no benefit to increasing polynomial degree.
• Figure 2: Active-set reduced-space iteration counts until the residuals or their relative reduction reach $10^{-8}$. Note that for the smooth solution the number of iterations required is stable as the mesh is refined, while in the nonsmooth case number iterations increase.
• Figure 3: Note that the standard SUPG solution exhibits oscillations which cause the violation of the analytical bounds on the solution, and that this issue is remedied by the use of the bound-preserving method.
• Figure 4: Visualisations of exact and numerical solutions to Example 3, §\ref{['sec:interior_layer']}, The numerical solution obtained from the variational inequality problem and discretised with BPSUPG using $\mathbb{P}_1$ elements, $h = 2^{-7}$, is shown in blue, while the regular SUPG finite element solution with the same discretisation parameters is shown in red. The dotted line is the exact solution, and grey horizontal lines indicate bounds on the exact solution which are known a priori.
• Figure 5: Visualisations of the exact solution to Example 4, given by \ref{['eq:u_3']}.

Theorems & Definitions (25)

• Proposition 2.1: Existence & uniqueness of solutions scott2022transport
• Remark 2.2: Further remarks on regularity
• Remark 2.3: Problems satisfying a priori bounds
• Remark 2.4: Preservation of the bound at the degrees of freedom
• Lemma 3.1: Consistency of the SUPG method
• proof
• Lemma 3.2: Continuity & coercivity properties of $a_h$
• proof
• Theorem 3.3: Best approximation
• proof