Balancing Discretization and Iteration Error in Finite Element A Posteriori Error Analysis

2013 ◽  
pp. 109-132
Author(s):  
Rolf Rannacher ◽  
Jevgeni Vihharev
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error Analysis ◽  
A Posteriori Error ◽  
Posteriori Error Analysis

Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality

Acta Numerica ◽  
2002 ◽  
Vol 11 ◽  
pp. 145-236 ◽  
Author(s):  
Michael B. Giles ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Posteriori Error ◽  
Integral Functionals ◽  
A Posteriori ◽  
A Posteriori Error Analysis ◽  
Adjoint Methods ◽  
A Posteriori Error ◽  
Recent Developments ◽  
Posteriori Error Analysis

We give an overview of recent developments concerning the use of adjoint methods in two areas: the a posteriori error analysis of finite element methods for the numerical solution of partial differential equations where the quantity of interest is a functional of the solution, and superconvergent extraction of integral functionals by postprocessing.

scholarly journals A Posteriori Error Analysis for a Lagrange Multiplier Method for a Stokes/Biot Fluid-Poroelastic Structure Interaction Model

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Houédanou Koffi Wilfrid
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Error Estimate ◽  
Posteriori Error ◽  
A Posteriori Error Estimate ◽  
Posteriori Error Estimate ◽  
A Posteriori ◽  
A Posteriori Error Analysis ◽  
A Posteriori Error ◽  
Posteriori Error Analysis

In this work, we develop an a posteriori error analysis of a conforming mixed finite element method for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium on isotropic meshes in ℝ d , d ∈ 2 , 3 . The approach utilizes a Lagrange multiplier method to impose weakly the interface conditions [Ilona Ambartsumyan et al., Numerische Mathematik, 140 (2): 513-553, 2018]. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, Helmholtz decomposition, and local approximation properties of the Clément interpolant. On the other hand, inverse inequalities and the localization technique based on simplexe-bubble and face-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

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