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

Abstract: 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.