Paper Info
Title
How accurate are finite elements on anisotropic triangulations in the maximum norm
Abstract
In Kopteva (2014) a counterexample of an anisotropic triangulation was given on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. This example was given in the context of a singularly perturbed reaction–diffusion equation. In this paper, we present further examples of unanticipated pointwise convergence behaviour of Lagrange finite elements on anisotropic triangulations. In particular, we show that linear finite elements may exhibit lower than expected orders of convergence for the Laplace equation, as well as for certain singular equations, and their accuracy depends not only on the linear interpolation error, but also on the mesh topology. Furthermore, we demonstrate that pointwise convergence rates which are worse than one might expect are also observed when higher-order finite elements are employed on anisotropic meshes. A theoretical justification will be given for some of the observed numerical phenomena.
Year
DOI
Venue
2020
10.1016/j.cam.2019.06.032
Journal of Computational and Applied Mathematics
Keywords
Field
DocType
Anisotropic triangulation,Maximum norm,Layer solutions,Anisotropic diffusion,Lagrange finite elements
Convergence (routing),Mathematical analysis,Laplace's equation,Finite element method,Triangulation (social science),Pointwise convergence,Linear interpolation,Counterexample,Mathematics,Pointwise
Journal
Volume
ISSN
Citations 
364
0377-0427
0
PageRank 
References 
Authors
0.34
0
1
Name
Order
Citations
PageRank
Natalia Kopteva113022.08