Uniform Hölder-norm bounds for finite element approximations of second-order elliptic equations
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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 -∇·(A∇ u)=f-∇· F with A∈ L^∞(Ω;R^n× n) a uniformly elliptic matrix-valued function, f∈ L^q(Ω), F∈ L^p(Ω;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 Ω⊂R^n.
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