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The classical arguments employed when obtaining error estimates of FiniteElement (FE) discretisations of elliptic problems lead to more restrictiveassumptions on the regularity of the exact solution when applied tonon-conforming methods. The so-called minimal regularity estimates available inthe literature relax some of these assumptions, but are not truly of -minimalregularity-, since a data oscillation term appears in the error estimate.Employing an approach based on a smoothing operator, we derive for the firsttime error estimates for Discontinuous Galerkin (DG) type discretisations ofnon-linear problems with $(p,\delta)$-structure that only assume the natural$W^{1,p}$-regularity of the exact solution, and which do not contain anyoscillation terms.

Quasi-optimal Discontinuous Galerkin discretisations of the $p$-Dirichlet problem