We propose a new method for the construction of layer-adapted meshes for singularly perturbed differential equations (SPDEs), based on mesh partial differential equations (MPDEs) that incorporate \emph{a posteriori} solution information. There are numerous studies on the development of parameter robust numerical methods for SPDEs that depend on the layer-adapted mesh of Bakhvalov. In~\citep{HiMa2021}, a novel MPDE-based approach for constructing a generalisation of these meshes was proposed. Like with most layer-adapted mesh methods, the algorithms in that article depended on detailed derivations of \emph{a priori} bounds on the SPDE's solution and its derivatives. In this work we extend that approach so that it instead uses \emph{a posteriori} computed estimates of the solution. We present detailed algorithms for the efficient implementation of the method, and numerical results for the robust solution of two-parameter reaction-convection-diffusion problems, in one and two dimensions. We also provide full FEniCS code for a one-dimensional example.

翻译：我们提出了一种基于网格偏微分方程（MPDE）并结合后验解信息的新型方法，用于构建奇异摄动微分方程（SPDE）的层适应网格。目前已有大量关于SPDE参数鲁棒数值方法的研究，这些方法依赖于Bakhvalov型层适应网格。在文献~\citep{HiMa2021}中，提出了一种基于MPDE的新方法，用于构造此类网格的推广形式。与大多数层适应网格方法类似，该文献中的算法依赖于对SPDE解及其导数的先验界进行详细推导。本文在现有方法基础上进行了拓展，改用后验计算得到的解估计值。我们给出了该方法高效实现的具体算法，并针对一维和二维双参数反应-对流-扩散问题的鲁棒求解提供了数值结果。此外，我们还提供了完整的一维示例FEniCS代码。