Convergence rate of the finite element approximation for extremizers of Sobolev inequalities

TL;DR

This paper analyzes how quickly finite element methods approximate extremal functions of Sobolev inequalities on convex polygonal domains, providing specific convergence rates in $L^2$ and $H^1$ norms.

Contribution

It establishes the convergence rates of FEM approximations to Sobolev extremizers on convex polygonal domains in $ extbf{R}^2$, with precise $O(h^2)$ and $O(h)$ bounds.

Findings

FEM solutions converge at $O(h^2)$ in $L^2$ norm.

FEM solutions converge at $O(h)$ in $H^1$ norm.

Results are valid for convex polygonal domains in $ extbf{R}^2$.

Abstract

In this paper, we are concerned with the convergence rate of a FEM based numerical scheme approximating extremal functions of the Sobolev inequality. We prove that when the domain is polygonal and convex in $\R^2$, the convergence of a finite element solution to an exact extremal function in $L^2$ and $H^1$ norms has the rates $O(h^2)$ and $O(h)$ respectively, where $h$ denotes the mesh size of a triangulation of the domain.