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

翻译：在获取椭圆问题有限元离散化误差估计时采用的经典论证方法，应用于非协调方法时会对精确解的正则性施加更严格的假设。文献中所谓的"最小正则性估计"放松了部分假设，但由于误差估计中存在数据振荡项，实际上并未达到真正的最小正则性要求。通过采用基于平滑算子的方法，我们首次为具有$(p,\delta)$结构的非线性问题的间断Galerkin型离散化推导出误差估计，该估计仅假设精确解具有自然的$W^{1,p}$正则性，且不包含任何振荡项。