Year: 2021
Author: Niall Madden, Róisín Hill, Niall Madden
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 3 : pp. 559–588
Abstract
We present a new algorithm for generating layer-adapted meshes for the finite element solution of singularly perturbed problems based on mesh partial differential equations (MPDEs). The ultimate goal is to design meshes that are similar to 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 is a nonlinear problem, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of mesh intervals. We resolve this by proposing a scheme based on $h$-refinement. We present fully working FEniCS codes [Alnaes et al., Arch. Numer. Softw., 3 (100) (2015)] that implement these methods, facilitating their extension to other problems and settings.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2020-0187
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 3 : pp. 559–588
Published online: 2021-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 30
Keywords: Mesh PDEs finite element method PDEs singularly perturbed layer-adapted meshes.
Author Details
-
Finite-dimensional perturbation of the Dirichlet boundary value problem for the biharmonic equation
Berikkhanova, Gulnaz
Zeitschrift für Naturforschung A, Vol. 79 (2024), Iss. 8 P.755
https://doi.org/10.1515/zna-2024-0020 [Citations: 0] -
Layer-adapted meshes for singularly perturbed problems via mesh partial differential equations and a posteriori information
Hill, Róisín | Madden, NiallComputers & Mathematics with Applications, Vol. 168 (2024), Iss. P.1
https://doi.org/10.1016/j.camwa.2024.05.019 [Citations: 0] -
A high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution
Kumar, Sunil | Vigo-Aguiar, JesusMathematics and Computers in Simulation, Vol. 199 (2022), Iss. P.287
https://doi.org/10.1016/j.matcom.2022.03.025 [Citations: 6]