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Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations [article]

Mark Ainsworth, Guosheng Fu
2017 arXiv   pre-print
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  ...  We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error.  ...  In comparison, there are relatively few works on a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods [22] .  ... 
arXiv:1706.05778v1 fatcat:abgikll7trgvnohkj52dscpehm

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

Mark Ainsworth, Guosheng Fu
2018 Journal of Scientific Computing  
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  ...  We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error.  ...  In comparison, there are relatively few works on a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods [22] .  ... 
doi:10.1007/s10915-018-0715-9 fatcat:7bcn25hz4jgrdhs4j3ayd73cqy

Hybridizable discontinuous Galerkin and mixed finite element methods for elliptic problems on surfaces

Bernardo Cockburn, Alan Demlow
2016 Mathematics of Computation  
Laplace-Beltrami operator, surface finite element methods, a priori error estimates, boundary value problems on surfaces, discontinuous Galerkin methods, hybridizable finite element methods, mixed finite  ...  We define and analyze hybridizable discontinuous Galerkin methods for the Laplace-Beltrami problem on implicitly defined surfaces.  ...  A hybridizable discontinuous Galerkin method In this section we define our hybridizable discontinuous Galerkin method. 3.1. Finite element spaces.  ... 
doi:10.1090/mcom/3093 fatcat:rqhnwcr3mvfdnll5ekpareaeha

A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method [article]

Nuria Pares, Ngoc-Cuong Nguyen, Pedro Diez, Jaume Peraire
2021 arXiv   pre-print
In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method.  ...  We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for  ...  Fu, Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations, J. Sci. Comput. 77 (1) (2018) 443-466.  ... 
arXiv:2106.10945v1 fatcat:ysvnnp7fp5e73bchh7wpexwklm

Nonstandard Finite Element Methods

Susanne Brenner, Carsten Carstensen, Peter Monk
2008 Oberwolfach Reports  
More recently, to improve robustness, computational aspects or to provide extra properties (e.g. appropriate conservation properties) new finite element methods such as discontinuous Galerkin methods,  ...  The mathematical analysis of standard conforming finite elements is very well advanced giving rise to highly efficient codes particularly for elliptic problems.  ...  The computational results seem very promising. We consider a further, more detailed analysis of the discretization method as well as the iterative solver is a challenging task.  ... 
doi:10.4171/owr/2008/36 fatcat:n524fqo2ojafngsmk3xef2vtlm

Frequency-explicit a posteriori error estimates for finite element discretizations of Maxwell's equations [article]

T. Chaumont-Frelet, P. Vega
2020 arXiv   pre-print
We consider residual-based a posteriori error estimators for Galerkin-type discretizations of time-harmonic Maxwell's equations.  ...  Our mathematical analysis is performed in the 3D case, and covers conforming N\'ed\'elec discretizations of the first and second family, as well as first-order (and hybridizable) discontinuous Galerkin  ...  We have focused on (conforming) Nédélec finite element discretizations of the second-order formulation, and first-order discontinuous Galerkin methods.  ... 
arXiv:2009.09204v1 fatcat:d3zw4rtusjbc3ieetxkyp4cauq

A Posteriori Error Estimates For Local C0 Discontinuous Galerkin Methods For Kirchhoff Plate Bending Problems

Yifeng Xu, Jianguo Huang, Xuehai Huang
2014 Journal of Computational Mathematics  
We derive some residual-type a posteriori error estimates for the local C 0 discontinuous Galerkin (LCDG) approximations ([31]) of the Kirchhoff bending plate clamped on the boundary.  ...  The estimator is both reliable and efficient with respect to the moment-field approximation error in an energy norm. Some numerical experiments are reported to demonstrate theoretical results.  ...  The authors thank the referees for their valuable comments leading to great improvement of the earlier version of the paper. This work was partially supported by the National Natural Science  ... 
doi:10.4208/jcm.1405-m4409 fatcat:lpsgyc566nhfpnmjb7ndbkiklu

AIR algebraic multigrid for a space-time hybridizable discontinuous Galerkin discretization of advection(-diffusion) [article]

Abdullah A. Sivas, Ben S. Southworth, Sander Rhebergen
2020 arXiv   pre-print
discontinuous Galerkin (HDG) discretization of advection-dominated flows.  ...  This paper investigates the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time hybridizable  ...  We are furthermore grateful for the computing resources provided by the Math Faculty Computing Facility at the University of Waterloo (https://uwaterloo.ca/math-faculty-computing-facility/).  ... 
arXiv:2010.11130v2 fatcat:nipeoeqyfvbapdm64vqvkoxsie

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

Verónica Anaya, David Mora, Ricardo Oyarzúa, Ricardo Ruiz-Baier
2015 Numerische Mathematik  
On the other hand, we also show that families of finite elements based on Brezzi-Douglas-Marini elements of order k +1 for the approximation of velocity, piecewise continuous polynomials of degree k +2  ...  We establish a priori error estimates in the natural norms with constants independent B Ricardo Ruiz-Baier  ...  We mention that all the error estimates are fully independent of the viscosity. Next, we develop a reliable and efficient residual-based a posteriori error estimator for the proposed formulation.  ... 
doi:10.1007/s00211-015-0758-x fatcat:q4cxwhjbtrf2lho5pn4moy46qi

hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity

Trevor Vincent, Harald P. Pfeiffer, Nils L. Fischer
2019 Physical Review D  
It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains.  ...  This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity.  ...  Computations were performed on the Minerva cluster at the Max-Planck-Institute for Gravitational Physics, and the GPC and Niagara supercomputers at the SciNet HPC Consortium [82] .  ... 
doi:10.1103/physrevd.100.084052 fatcat:q24fsl36tzctnpyftczixuhuzy

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

Matteo Giacomini, Ruben Sevilla
2019 SN Applied Sciences  
Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method.  ...  Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes.  ...  On the one hand, the flexibility of DG methods has been exploited to perform mesh refinement based on octrees [78] , driven by adjointbased [79] and fully-computable [80] a posteriori error estimators  ... 
doi:10.1007/s42452-019-1065-4 fatcat:bk6m73k4rfdzlgmloz7cibwmsm

Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations [article]

Xudong Wang, Weihua Deng
2017 arXiv   pre-print
This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations.  ...  Based on the derived posteriori error estimates, the local error indicator is designed.  ...  Let u ∈ V be the solution of (3.27) ∂ ∂t u, v + a(u, v) − κ(u, v) = (f, v) ∀v ∈ V, and u n h be the fully discrete discontinuous finite element approximation defined by (3.28) (∂ t u n h , v h ) + a(u  ... 
arXiv:1706.02826v1 fatcat:mbnkwg4nnrdefovksfkos67uom

Slope limiting the velocity field in a discontinuous Galerkin divergence-free two-phase flow solver

Tormod Landet, Kent-Andre Mardal, Mikael Mortensen
2019 Computers & Fluids  
This thesis describes a finite element method for simulation of free-surface flows, such as ocean waves, using the discontinuous Galerkin method.  ...  This thesis shows how slope limiting can be used to stabilise the momentum equation in an exactly mass conserving discontinuous Galerkin finite element discretisation of the Navier-Stokes equation.  ...  "An interior penalty finite element method with discontinuous elements". SIAM journal on numerical analysis 19.4, pp. 742-760. Baker, G. A., Jureidini, W. N., and Karakashian, O. A. (Dec. 1990).  ... 
doi:10.1016/j.compfluid.2019.104322 fatcat:b7ggj2clvjethksh7k5yddc2wq

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems

Bernardo Cockburn, Jayadeep Gopalakrishnan, Raytcho Lazarov
2009 SIAM Journal on Numerical Analysis  
The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continuous Galerkin, non-conforming and a new, wide class of hybridizable discontinuous  ...  We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems.  ...  The first author would like to thank Martin Vohralík for bringing to his attention the reference [21] .  ... 
doi:10.1137/070706616 fatcat:kjt3gcdezjc2lktmruu6mkp5ka

Meshless techniques for anisotropic diffusion

Annamaria Mazzia, Giorgio Pini, Flavio Sartoretto
2014 Applied Mathematics and Computation  
The latter would allow better use of parallel computers, since time-stepping is essentially a serial process. Moreover, it would be good for the methods to be of high order accuracy.  ...  A good numerical method would be locally mass conservative, produce no or minimal over/under-shoots, produce minimal numerical diffusion, and require no CFL time-step limit for stability.  ...  We show how to solve the equations using a global implicit approach in an efficient way, and we present the derived computational results.  ... 
doi:10.1016/j.amc.2014.03.032 fatcat:c527226gyfgbffnq4p67qxd7wi
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