A posteriori error analysis and adaptive processes in the finite element method: Part II—adaptive mesh refinement

1983 ◽  
Vol 19 (11) ◽  
pp. 1621-1656 ◽  
Author(s):  
J. P. De S. R. Gago ◽  
D. W. Kelly ◽  
O. C. Zienkiewicz ◽  
I. Babuska
Keyword(s):  
Finite Element Method ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error Analysis ◽  
The Finite Element Method ◽  
A Posteriori Error ◽  
Adaptive Processes ◽  
Posteriori Error Analysis

scholarly journals Residual-based a posteriori error analysis for the coupling of the Navier-Stokes and Darcy-Forchheimer equations

2021 ◽  
Author(s):  
Sergio Caucao ◽  
Gabriel Gatica ◽  
Ricardo Oyarzúa ◽  
Felipe Sandoval
Keyword(s):  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Posteriori Error ◽  
Local Approximation ◽  
Navier Stokes ◽  
Error Estimator ◽  
A Posteriori ◽  
Approximation Properties ◽  
A Posteriori Error ◽  
Posteriori Error Analysis

In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier--Stokes and Darcy--Forchheimer equations, and derive,  though in a non-standard sense,  a reliable and efficient residual-based a posteriori error estimator suitable for an adaptive mesh-refinement method.  For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium, local approximation properties of the Cl\'ement interpolant and Raviart--Thomas operator, and a smallness assumption on the data.   In turn, inverse inequalities, the localization technique based on triangle-bubble and edge-bubble functions in local $\L^\rp$ spaces, are the main tools for developing the effi\-ciency analysis, which is valid for the error estimator itself up to a suitable additional error term. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

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