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

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 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 [R. Hill, N. Madden, Number. Math. Theory Methods Appl. 14, 559–588], 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 a priori bounds on the SPDE’s solution and its derivatives. In this work we extend that approach so that it instead uses 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.