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

• Main Authors: Hill, Róisín, Madden, Niall
• Format: Journal Article
• Language: English
• Published: 02-11-2023
• Subjects: Computer Science - Numerical Analysis; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.2311.01274

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.