On optimal $L^2$- and surface flux convergence in FEM (extended version)

TL;DR

This paper establishes conditions under which optimal $L^2$-convergence rates are achieved in finite element methods, considering boundary regularity and domain shape, with implications for boundary flux approximation.

Contribution

It provides new convergence results for FEM under weaker regularity assumptions and analyzes boundary flux approximation without convexity constraints.

Findings

Optimal $L^2$-convergence is achievable under certain boundary regularity conditions.

Boundary flux approximation attains near-optimal rates without convexity assumptions.

Regularity of the solution influences the convergence rates in FEM.

Abstract

We show that optimal $L^2$-convergence in the finite element method on quasi-uniform meshes can be achieved if, for some $s_0 > 1/2$, the boundary value problem has the mapping property $H^{-1+s} \rightarrow H^{1+s}$ for $s \in [0,s_0]$. The lack of full elliptic regularity in the dual problem has to be compensated by additional regularity of the exact solution. Furthermore, we analyze for a Dirichlet problem the approximation of the normal derivative on the boundary without convexity assumption on the domain. We show that (up to logarithmic factors) the optimal rate is obtained.