This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of $O(h^k)$ in the energy norm, where $k$ represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach.

本文介绍了一种新颖的方法，使用一致的有限元方法来近似各种反应-对流-扩散方程，同时提供尊重底层微分方程给出的物理边界的离散解。这项工作的主要结果表明，数值解在能量范数中达到了 $O(h^k)$ 的精度，其中 $k$ 表示基础多项式次数。为了验证该方法，针对各种问题实例进行了一系列数值实验。与线性连续内罚稳定方法和代数通量校正方案（针对分段线性有限元情况）进行了比较，我们可以观察到当前方法的良好性能。