An a posteriori analysis of some inconsistent, nonconforming Galerkin methods approximating elliptic problems

In this work we present an a posteriori analysis for classes of inconsistent, nonconforming schemes approximating elliptic problems. We show the estimates coincide with existing ones for interior penalty type discontinuous Galerkin approximations of the Laplacian and give new estimates for inconsistent discontinuous Galerkin approximation schemes of elliptic problems under quadrature approximation. We also examine the effect of inconsistencies on the a posteriori analysis of schemes applied to an unbalanced problem.