Uniform Hölder-norm bounds for finite element approximations of second-order elliptic equations
Lars Diening, Toni Scharle, ENDRE E. SÜLI · IMA Journal of Numerical Analysis · 2021
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 $- abla \cdot (A abla u)=f- abla \cdot F$ 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 $\varOmega \subset {{\mathbb{R}}}^n$.