This paper presents a modified graded mesh for singularly perturbed reaction–diffusion problems. The mesh we offer is generated recursively using Newton’s algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on polynomials of degree $p\ge 1$. We prove parameter uniform convergence of optimal order in $ϵ$-weighted energy norm. Test examples are taken, and we present a rigorous comparative analysis with other adaptive meshes. Moreover, we compare the proposed method with others found in the literature.

中文翻译：

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奇异摄动反应扩散问题的改进梯度网格和高阶有限元方法

本文提出了一种用于奇异摄动反应扩散问题的改进的渐变网格。我们提供的网格是使用牛顿算法和一些隐式定义的函数递归生成的。使用基于度多项式的有限元方法以数值方式解决该问题$p\ge \mathrm{1个}$。我们证明了最优阶的参数一致收敛$ϵ$加权能源规范。以测试示例为例，我们提出了与其他自适应网格的严格比较分析。此外，我们将提出的方法与文献中发现的其他方法进行了比较。