A posteriori error analysis of hp-version discontinuous Galerkin finite-element methods for second-order quasi-linear elliptic PDEs

We develop the a posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite-element methods for a class of second-order quasi-linear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh dependent) energy norm. The bounds are explicit in the local mesh size and the local polynomial degree of the approximating finite element function. The performance of the proposed error indicators within an automatic hp-adaptive refinement procedure is studied through numerical experiments.