Uniform Hölder-norm bounds for finite element approximations of second-order elliptic equations

Abstract

We develop a discrete counterpart of the De Giorgi-Nash-Moser theory, which provides uniform Hölder-norm bounds on continuous piecewise affine finite element approximations of second-order linear elliptic problems of the form $-\nabla \cdot (A\nabla u)=f-\nabla \cdot F$ with $A\in L^{\infty }(\Omega ;{\mathbb{R}}^{n\times n})$ a uniformly elliptic matrix-valued function, $f\in L^q(\Omega )$, $F\in L^p(\Omega ;{\mathbb{R}}^n)$, with $p>n$ and $q>n/2$, on $A$-nonobtuse shape-regular triangulations, which are not required to be quasi-uniform, of a bounded polyhedral Lipschitz domain $\Omega \subset {\mathbb{R}}^n$.

The paper has been accepted for publication in the IMAJNA on 24th March 2021