Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations

We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial meshes in two and three dimensions. We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error. The estimators are also shown to provide local lower bounds for the HDG energy (semi)norm of the error up to a constant and a higher-order data oscillation term. For the primal HDG methods and mixed HDG methods with an appropriate choice of stabilization parameter, the estimators are also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and a higher-order data oscillation term. Numerical examples are given illustrating the theoretical results

An embedded--hybridized discontinuous Galerkin finite element method for the Stokes equations

We present and analyze a new embedded--hybridized discontinuous Galerkin finite element method for the Stokes problem. The method has the attractive properties of full hybridized methods, namely an $H({\rm div})$-conforming velocity field, pointwise satisfaction of the continuity equation and \emph{a priori} error estimates for the velocity that are independent of the pressure. The embedded--hybridized formulation has advantages over a full hybridized formulation in that it has fewer global degrees-of-freedom for a given mesh and the algebraic structure of the resulting linear system is better suited to fast iterative solvers. The analysis results are supported by a range of numerical examples that demonstrate rates of convergence, and which show computational efficiency gains over a full hybridized formulation