Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems (Q607692)

The author investigates a general second-order elliptic boundary value problem in two dimensions and two variants of the discontinuous Galerkin method (mainly the symmetric but also the non-symmetric interior penalty method) for its numerical solution. Here, earlier results assume a \(u\in H^{1+\alpha}\) regularity along with \(1\geq\alpha>1/2\) whereas the author is able to achieve his a priori error estimates supposing \(1\geq\alpha>0\) only. For this, he proves a Garding type inequality for the discrete problem and uses techniques from a posteriori error estimates as developed by Verfürth to show unique solvability of the adjoint discrete problem. Further he obtains his error estimates in a discrete \(H^1\)-norm and, finally, optimal estimates in the \(L_2\)-norm for the symmetric interior penalty method.