Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

2002/01/01 by Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn +2 · 117 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #Numerical methods for differential equations

paper · doi:10.1137/s0036142901384162

Abstract

Abstract. We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

Citations

• An Interior Penalty Finite Element Method with Discontinuous Elements
• Mixed finite elements in ?3
• A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier–Stokes Equations
• The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws V

Cited by

• The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems
• A Local Macroscopic Conservative Low-Rank Discontinuous Galerkin Method for the Vlasov-Poisson Equation with Dougherty-Fokker-Planck Collisions
• An agglomeration-based multigrid solver for the discontinuous Galerkin discretization of cardiac electrophysiology
• A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation
• Implicit multistage two-derivative discontinuous Galerkin schemes for\n viscous conservation laws
• A Reconstructed Discontinuous Approximation to Monge-Ampere Equation in Least Square Formation
• On stabilizer-free weak Galerkin finite element methods on polytopal meshes
• On the discontinuous Galerkin method for solving boundary value problems for the Helmholtz equation: A priori and a posteriori error analyses
• A total variation discontinuous Galerkin approach for image restoration
• Turbulent jet simulation using high-order DG methods for aeroacoustics analysis
• A postprocessing technique for a discontinuous Galerkin discretization of time-dependent Maxwell's equations
• Analysis of Discontinuous Galerkin Methods using Mesh-Dependent Norms\n and Applications to Problems with Rough Data
• Discontinuous Galerkin methods for semilinear elliptic boundary value problem
• From Raviart-Thomas to HDG
• A Least Squares Weak Galerkin Framework for Linear Elasticity on Polytopal Meshes
• The analysis of FETI-DP preconditioner for full DG discretization of elliptic problems
• A massively parallel non-overlapping Schwarz preconditioner for PolyDG methods in brain electrophysiology
• A divergence-free parametric finite element method for 3D Stokes equations on curved domains
• On the Stability of Continuous-Discontinuous Galerkin Methods for\n Advection-Diffusion-Reaction Problems
• A Discontinuous Galerkin Consistent Splitting Method for the Incompressible Navier-Stokes Equations
• Divergence-free decoupled finite element methods for incompressible flow problems
• A Nitsche Hybrid multiscale method with non-matching grids
• Discontinuous Galerkin methods for the p--biharmonic equation from a\n discrete variational perspective
• Can Explicit Subgrid Models Enhance Implicit LES Simulations? A GPU-Oriented High-Order-Solver Perspective
• The discontinuous Galerkin method for the Oseen eigenvalue problem
• A unified error analysis of HDG methods for the static Maxwell equations
• 3D Modeling of Solar Oscillations with Hybridizable Discontinuous Galerkin Method
• A finite element method for a non-Newtonian dilute polymer fluid
• A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry
• Discontinuous Galerkin methods for nonvariational problems
• A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods
• Analysis of a new stabilized discontinuous Galerkin method for the reaction-diffusion problem with discontinuous coefficient
• Superconvergence of Discontinuous Galerkin methods for Elliptic Boundary Value Problems
• On discontinuous Galerkin discretizations of second-order derivatives
• V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes
• Explicit Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element
• Stability estimates for Interior Penalty D.G. Methods for the Nonlinear Dynamics of the complex Ginzburg Landau equation
• Numerical boundary flux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries
• Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Equations
• High Order Semi-Lagrangian Discontinuous Galerkin Method Coupled with Runge-Kutta Exponential Integrators for Nonlinear Vlasov Dynamics
• A conforming discontinuous Galerkin finite element method
• HORSES3D-GPU: A high-order discontinuous Galerkin solver for multi-GPU systems
• A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods
• An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids
• Modern discontinuous Galerkin methods for the simulation of transitional\n and turbulent flows in biomedical engineering: A comprehensive LES study of\n the FDA benchmark nozzle model
• A high order semi-Lagrangian discontinuous Galerkin method for the two-dimensional incompressible Euler equations and the guiding center Vlasov model without operator splitting
• Anisotropic dynamic models for Large Eddy Simulation of compressible flows with a high order DG method
• Weighted Relaxation for Multigrid Reduction in Time
• On mathematical and numerical modelling of multiphysics wave propagation\n with polytopal Discontinuous Galerkin methods
• Discontinuous Galerkin methods for the Laplace-Beltrami operator on point cloud
• Multi-dimensional filtering: Reducing the dimension through rotation
• Meshless discretization of LQ-type stochastic control problems
• Comparative study of finite element methods using the Time-Accuracy-Size (TAS) spectrum analysis
• Analysis of a family of HDG methods for second order elliptic problems
• Optimization of two-level methods for DG discretizations of reaction-diffusion equations
• Direct and Large Eddy Simulation of three-dimensional non-Boussinesq\n gravity currents with a high order DG method
• An Energy-Stable Discontinuous Galerkin Method for the Compressible Navier--Stokes--Allen--Cahn System
• Fast adaptive discontinuous basis sets for electronic structure
• BPX preconditioner for nonstandard finite element methods for diffusion problems
• Optimal L² error estimation for the unfitted interface finite element method based on the non-symmetric Nitsche's methods
• Divergence-conforming methods for transient doubly-diffusive flows: A priori and a posteriori error analysis
• A posteriori analysis for nonlinear convection-diffusion systems
• A Priori hp-estimates for Discontinuous Galerkin Approximations to\n Linear Hyperbolic Integro-Differential Equations
• DSA Preconditioning for DG discretizations of S_N transport and High-Order curved meshes
• Optimal L² Error Estimates for Non-symmetric Nitsche's Methods
• High order well-balanced and total-energy-conserving local discontinuous Galerkin methods for compressible self-gravitating Euler equations
• Comparison of semi-Lagrangian discontinuous Galerkin schemes for linear and nonlinear transport simulations
• Deep Learning Accelerated Algebraic Multigrid Methods for Polytopal Discretizations of Second-Order Differential Problems
• FESTIM v2.0: Upgraded framework for multi-species hydrogen transport and enhanced performance
• Shape Optimization using the Finite Element Method on Multiple Meshes
• Mathematical and numerical modeling of coupled oxygen dynamics and neuronal electrophysiology
• The effects of maternal flow on placental diffusion‐weighted MRI and intravoxel incoherent motion parameters
• Algebraic Multigrid Preconditioning of High-Order Spectral Elements for Elliptic Problems on a Simplicial Mesh
• Finite volume schemes of any order on rectangular meshes
• A Combined Preconditioning Strategy for Nonsymmetric Systems
• C⁰ Discontinuous Galerkin Methods for a Kirchhoff Plate Contact Problem
• Energy norm error estimates for averaged discontinuous Galerkin methods:\n multidimensional case
• An adaptive high-order unfitted finite element method for elliptic interface problems
• Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshes
• An Analysis of the Weak Finite Element Method for Convection-Diffusion Equations
• p-multigrid method for the discontinuous Galerkin discretization of elliptic problems
• Linear and Nonlinear Boundary Conditions: What's the difference?
• Locally conservative immersed finite element method for elliptic interface problems
• Unconditionally stable second order convergent partitioned methods for multiple-network poroelasticity
• Reduced basis solvers for unfitted methods on parameterized domains
• Conforming and discontinuous discretizations of non-isothermal Darcy-Forchheimer flows
• An Extended Galerkin Analysis for Elliptic Problems
• Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs
• \boldsymbolH(curl)-reconstruction of piecewise polynomial fields with application to hp-a posteriori nonconforming error analysis for Maxwell's equations
• An hp-Adaptive Newton-Discontinuous-Galerkin Finite Element Approach\n for Semilinear Elliptic Boundary Value Problems
• A Unified Study of Continuous and Discontinuous Galerkin Methods
• Weak Galerkin Finite Element Methods on Polytopal Meshes
• A family of three-dimensional virtual elements with applications to magnetostatic
• Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework II: Force, vibration, and molecular dynamics calculations
• Residual-based a posteriori error estimation for immersed finite element methods
• A Weak Galerkin Mixed Finite Element Method for Second-Order Elliptic Problems
• Discontinuous Galerkin approximation of linear parabolic problems with\n dynamic boundary conditions
• Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes
• A conforming discontinuous Galerkin finite element method: Part II
• An entropy-stable and kinetic energy-preserving macro-element HDG method for compressible flows
• Implicit multiderivative collocation solvers for linear partial\n differential equations with discontinuous Galerkin spatial discretizations
• A sequential least squares method for elliptic equations in non-divergence form
• A stabilized Nitsche overlapping mesh method for the Stokes problem
• On the Penalty term for the Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation
• Polynomial robust stability analysis for H(\textrmdiv)-conforming finite elements for the Stokes equations
• Nitsche's method for a Robin boundary value problem in a smooth domain
• A structure-preserving discontinuous Galerkin scheme for the Fischer-KPP equation
• Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulation
• Hybridized discontinuous Galerkin method for convection-diffusion problems
• MVP-Workshop Contribution: Modeling of Volvo Bluff Flame Experiment and Comparison of Finite-Volume and Discontinuous-Galerkin Schemes
• An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid
• A stabilized finite element method for delamination analysis of\n composites using cohesive elements
• Adaptively enriched coarse space for the discontinuous Galerkin\n multiscale problems
• Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods
• Discrete Korn's inequality for shells
• Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number
• A conservative local multiscale model reduction technique for Stokes flows in heterogeneous perforated domains

Related

• An Extended Galerkin Analysis for Elliptic Problems
• Discontinuous Galerkin methods for nonvariational problems
• An Optimal Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems
• An a posteriori analysis of some inconsistent, nonconforming Galerkin methods approximating elliptic problems
• High order discontinuous Galerkin methods on surfaces