2018/11/13 by Natalia Kopteva, Kopteva, Natalia
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1811.05353
openalex publication_date 2018/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In [Kopteva, Math. Comp., 2014] a counterexample of an anisotropic\ntriangulation was given on which the exact solution has a second-order error of\nlinear interpolation, while the computed solution obtained using linear finite\nelements is only first-order pointwise accurate. This example was given in the\ncontext of a singularly perturbed reaction-diffusion equation. In this paper,\nwe present further examples of unanticipated pointwise convergence behaviour of\nLagrange finite elements on anisotropic triangulations. In particular, we show\nthat linear finite elements may exhibit lower than expected orders of\nconvergence for the Laplace equation, as well as for certain singular\nequations, and their accuracy depends not only on the linear interpolation\nerror, but also on the mesh topology. Furthermore, we demonstrate that\npointwise convergence rates which are worse than one might expect are also\nobserved when higher-order finite elements are employed on anisotropic meshes.\nA theoretical justification will be given for some of the observed numerical\nphenomena.\n