Volume 14, Issue 3
Generating Layer-Adapted Meshes Using Mesh Partial Differential Equations

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 559-588.

Published online: 2021-06

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• 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.

• Keywords

Mesh PDEs, finite element method, PDEs, singularly perturbed, layer-adapted meshes.

65N30, 65N50, 65-04

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@Article{NMTMA-14-559, author = {Hill , Róisín and Madden , Niall}, title = {Generating Layer-Adapted Meshes Using Mesh Partial Differential Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {3}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0187}, url = {http://global-sci.org/intro/article_detail/nmtma/19189.html} }
TY - JOUR T1 - Generating Layer-Adapted Meshes Using Mesh Partial Differential Equations AU - Hill , Róisín AU - Madden , Niall JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 559 EP - 588 PY - 2021 DA - 2021/06 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0187 UR - https://global-sci.org/intro/article_detail/nmtma/19189.html KW - Mesh PDEs, finite element method, PDEs, singularly perturbed, layer-adapted meshes. AB -

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.

Róisín Hill & Niall Madden. (2021). Generating Layer-Adapted Meshes Using Mesh Partial Differential Equations. Numerical Mathematics: Theory, Methods and Applications. 14 (3). 559-588. doi:10.4208/nmtma.OA-2020-0187
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