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

Lars Diening, Toni Scharle, Endre Süli
2021 IMA Journal of Numerical Analysis  
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
more » ... -regular triangulations, which are not required to be quasi-uniform, of a bounded polyhedral Lipschitz domain $\varOmega \subset {{\mathbb{R}}}^n$.
doi:10.1093/imanum/drab029 fatcat:bzcwazh4rzbebemtw4e2ishcza