| Abstract: | We present a new algorithm for generating layer-adapted meshes for thefinite element solution of singularly perturbed problems based on mesh partial differential equations (MPDEs). The ultimate goal is to design meshes that are similarto the well-known Bakhvalov meshes, but can be used in more general settings:specifically two-dimensional problems for which the optimal mesh is not tensor-product in nature. Our focus is on the efficient implementation of these algorithms,and numerical verification of their properties in a variety of settings. The MPDE isa nonlinear problem, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of meshintervals. We resolve this by proposing a scheme based on $h$-refinement. We presentfully working FEniCS codes [Alnaes et al., Arch. Numer. Softw., 3 (100) (2015)]that implement these methods, facilitating their extension to other problems andsettings. |
