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

## TL;DR

This paper develops fully computable a posteriori error bounds for hybridizable discontinuous Galerkin methods applied to second-order elliptic problems, enabling reliable error estimation in finite element approximations.

## Contribution

It introduces constant-free, fully computable a posteriori error estimators for primal and mixed HDG methods, including local lower bounds and applicability to various formulations.

## Key findings

- Error bounds are fully computable and constant-free.
- Estimators provide local lower bounds up to a constant.
- Numerical examples confirm theoretical accuracy.

## 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.