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

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).

arXiv comment29 pages, 7 figures

References in corpus (23)

• A Hybrid High-Order method for the incompressible Navier--Stokes equations based on Temam's device
• High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows
• A hybridizable discontinuous Galerkin method for the Navier--Stokes equations with pointwise divergence-free velocity field
• A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers
• A Hybrid High-Order method for the steady incompressible Navier--Stokes problem
• A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow
• A hybridizable discontinuous Galerkin method for solving nonlocal optical response models
• Hybrid High-Order methods for finite deformations of hyperelastic materials
• Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods
• A super--convergent hybridisable discontinuous Galerkin method for linear elasticity
• A hybridizable discontinuous Galerkin method for computing nonlocal electromagnetic effects in three-dimensional metallic nanostructures
• A face-centred finite volume method for second-order elliptic problems
• A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor
• HDG-NEFEM with degree adaptivity for Stokes flows
• A locking-free face-centred finite volume (FCFV) method for linear elasticity
• A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows
• A Hybrid High-Order method for incremental associative plasticity with small deformations
• Computing parametrized solutions for plasmonic nanogap structures
• A Hybridizable Discontinuous Galerkin solver for the Grad-Shafranov equation
• Hybrid coupling of CG and HDG discretizations based on Nitsche's method
• An Improved Iterative HDG Approach for Partial Differential Equations
• Hybrid discontinuous Galerkin discretisation and domain decomposition preconditioners for the Stokes problem
• A Divergence-Conforming Hybridized Discontinuous Galerkin Method for the Incompressible Reynolds Averaged Navier-Stokes Equations

Cited by in corpus (8)

• Hybridisable discontinuous Galerkin formulation of compressible flows
• HDGlab: An open-source implementation of the hybridisable discontinuous Galerkin method in MATLAB
• Tutorial on Hybridizable Discontinuous Galerkin (HDG) Formulation for Incompressible Flow Problems
• Hybridisable discontinuous Galerkin solution of geometrically parametrised Stokes flows
• Hybrid coupling of CG and HDG discretizations based on Nitsche's method
• A non-oscillatory face-centred finite volume method for compressible flows
• An unfitted high-order HDG method for two-fluid Stokes flow with exact NURBS geometries
• Unified $hp$-HDG Frameworks for Friedrichs' PDE systems