# Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

## TL;DR

This paper revisits Discontinuous Galerkin methods with exact geometry, degree adaptivity, and hybridization, demonstrating their effectiveness in complex simulations like electrostatics and fluid flows with improved accuracy and efficiency.

## Contribution

It introduces a hybridizable DG method with exact geometry representation and degree adaptivity, enhancing computational efficiency and convergence in practical applications.

## Key findings

- HDG achieves optimal convergence and superconvergence properties.
- Error indicators effectively guide degree adaptive procedures.
- Applications demonstrate improved accuracy in electrostatics, elasticity, and fluid flow simulations.

## Abstract

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).