Analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes

The equilibrated residual method is now accepted as the best residual type a posteriori error estimator. Nevertheless, there remains a gap in the theory and practice of the method. The present work tackles the problem of existence, construction and stability of equilibrated fluxes for hp-finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements, with hanging nodes. A practical algorithm for post-processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general, one-irregular partition. The resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh-size. Numerical examples are included to illustrate the theoretical results.