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

Layer-adapted meshes for singularly perturbed problems via mesh partial differential equations and a posteriori information