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

• Published in: IMA journal of numerical analysis 2021-07, Vol.41 (3), p.1846-1898
• Main Authors: Diening, Lars, Scharle, Toni, Süli, Endre
• Format: Article
• Language: English
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/drab029

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 $A\in L^\infty (\varOmega ; {{\mathbb{R}}}^{n\times n})$ a uniformly elliptic matrix-valued function, $f\in L^{q}(\varOmega )$, $F\in L^p(\varOmega ; {{\mathbb{R}}}^n)$, 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 $\varOmega \subset {{\mathbb{R}}}^n$.