Fully Computable a Posteriori Error Bounds for Hybridizable Discontinuous Galerkin Finite Element Approximations

Abstract

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.

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…Starting from the seminal works [104,105] establishing reliability and efficiency of error estimates for the HDG approximations of second-order elliptic equations, a posteriori estimates were developed for steady and unsteady scalar convection-diffusion problems [57,182] and for the vectorial case of incompressible Oseen [23] and Brinkman [22] flows. In addition, constant-free computable a posteriori error estimates are devised in [19] for second-order elliptic problems using an equilibrated fluxes approach, whereas residual-based estimates are established for Maxwell's equations in [58].…”

A two-energies principle for the biharmonic equation and ana posteriorierror estimator for an interior penalty discontinuous Galerkin approximation

“…The moment tensor p h can be evaluated by solving the small linear system (4.11a) for each K 2 T h . u (1) ,û (2) , b p and b , be given by (4.10) and suppose that the penalty parameters ↵ i , 1  i  2, are su ciently large.…”

“…Proof. The interpolation conditions for p (1) and p (2) are separated. The vector field p (i) (for 1  i  2) is determined by the degrees of freedom Z E p (i) n E q ds, q 2 P`(E), E 2 E h (@K).,…”

“…It was originally developed by Prager and Synge [36,38,39] and more recently considered in connection with second order elliptic problems in [1,7,8,9,10,11,12,41]. The considerations of DG methods in this direction [2,3,18,22,23,24,25] were also done for equations of second order.…”

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

“…Cockburn et al studied the contraction property of adaptive HDG methods for Poisson problem with homogeneous Dirichlet condition. Recently, Ainsworth and Fu derived a posteriori error estimates for the HDG methods for the approximation of a linear second‐order elliptic problem on conforming simplicial meshes in two and three dimensions. For quadrilateral meshes, Rivière and Wheeler gave a posteriori error estimates for a DG method applied to elliptic problems.…”