scholarly journals A simple virtual element-based flux recovery on quadtree

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Shuhao Cao
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Ad Hoc ◽  
Error Estimator ◽  
A Posteriori ◽  
Hanging Nodes ◽  
Virtual Element

<p style='text-indent:20px;'>In this paper, we introduce a simple local flux recovery for <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{Q}_k $\end{document}</tex-math></inline-formula> finite element of a scalar coefficient diffusion equation on quadtree meshes, with no restriction on the irregularities of hanging nodes. The construction requires no specific ad hoc tweaking for hanging nodes on <inline-formula><tex-math id="M2">\begin{document}$ l $\end{document}</tex-math></inline-formula>-irregular (<inline-formula><tex-math id="M3">\begin{document}$ l\geq 2 $\end{document}</tex-math></inline-formula>) meshes thanks to the adoption of virtual element families. The rectangular elements with hanging nodes are treated as polygons as in the flux recovery context. An efficient <i>a posteriori</i> error estimator is then constructed based on the recovered flux, and its reliability is proved under common assumptions, both of which are further verified in numerics.</p>

A POSTERIORI ERROR ESTIMATORS VIA BUBBLE FUNCTIONS

1996 ◽  
Vol 06 (01) ◽  
pp. 33-41 ◽  
Author(s):  
ALESSANDRO RUSSO
Keyword(s):  
Finite Element ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimators ◽  
Element Space ◽  
A Posteriori Error ◽  
Error Estimators ◽  
Bubble Functions ◽  
Posteriori Error Estimators

In this paper we discuss a way to recover a classical residual-based error estimator for elliptic problems by using a finite element space enriched with bubble functions. The advection-dominated case is also discussed.

scholarly journals Residual-based a posteriori error estimation for hp-adaptive finite element methods for the Stokes equations

2019 ◽  
Vol 27 (4) ◽  
pp. 237-252
Author(s):  
Arezou Ghesmati ◽  
Wolfgang Bangerth ◽  
Bruno Turcksin
Keyword(s):  
Finite Element ◽  
Error Estimation ◽  
Posteriori Error ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
Posteriori Error Estimation ◽  
A Posteriori Error Estimator ◽  
Posteriori Error Estimator ◽  
A Posteriori Error

AbstractWe derive a residual-based a posteriori error estimator for the conforminghp-Adaptive Finite Element Method (hp-AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classicalh-version of AFEM. We also establish the reliability and efficiency of the error estimator. The proofs are based on the well-known Clément-type interpolation operator introduced in [27] in the context of thehp-AFEM. Numerical experiments show the performance of an adaptivehp-FEM algorithm using the proposed a posteriori error estimator.

A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem

2017 ◽  
Vol 7 (3) ◽  
pp. 508-529 ◽  
Author(s):  
Xiaobo Zheng ◽  
Xiaoping Xie
Keyword(s):  
Finite Element ◽  
Lower Bounds ◽  
Stokes Problem ◽  
Three Dimensions ◽  
Upper And Lower Bounds ◽  
Error Estimator ◽  
A Posteriori ◽  
Weak Galerkin ◽  
Galerkin Finite Element ◽  
Velocity Approximation

AbstractA robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the L2-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

scholarly journals RESIDUAL-BASED A POSTERIORI ESTIMATORS FOR THE A - φ MAGNETODYNAMIC HARMONIC FORMULATION OF THE MAXWELL SYSTEM

2012 ◽  
Vol 22 (05) ◽  
pp. 1150028 ◽  
Author(s):  
EMMANUEL CREUSÉ ◽  
SERGE NICAISE ◽  
ZUQI TANG ◽  
YVONNICK LE MENACH ◽  
NICOLAS NEMITZ ◽  
...  
Keyword(s):  
Ad Hoc ◽  
Posteriori Error ◽  
Error Estimator ◽  
Maxwell System ◽  
A Posteriori ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Numerical Tests ◽  
Analytical Tools ◽  
Theoretical Results

This paper is devoted to the derivation of an a posteriori residual-based error estimator for the A-φ magnetodynamic harmonic formulation of the Maxwell system. The weak continuous and discrete formulations are established, and the well-posedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad hoc Helmholtz decomposition is proven, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results.

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