An A Posteriori Error Estimator for Adaptive Mesh Refinement Using Parallel In-Element Particle Tracking Methods

2003 ◽  
pp. 94-107 ◽  
Author(s):  
Jing-Ru C. Cheng ◽  
Paul E. Plassmann
Keyword(s):  
Particle Tracking ◽  
Adaptive Mesh Refinement ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Posteriori Error Estimator ◽  
A Posteriori Error

scholarly journals Flux recovery for Cut finite element method and its application in a posteriori error estimation

2021 ◽  
Author(s):  
Daniela Capatina ◽  
Cuiyu He
Keyword(s):  
Finite Element ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Posteriori Error ◽  
Numerical Results ◽  
Error Estimator ◽  
A Posteriori ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
The Difference

In this article, we aim to recover locally conservative and $H(div)$ conforming fluxes for the linear  Cut Finite Element Solution with Nitsche's method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The $L^2$-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we also prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.

An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem

2021 ◽  
Vol 36 (6) ◽  
pp. 313-336
Author(s):  
Ronald H. W. Hoppe ◽  
Youri Iliash
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Interior Penalty ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Flux Function

Abstract We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

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