scholarly journals A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems

2003 ◽  
Vol 13 (08) ◽  
pp. 1219-1229 ◽  
Author(s):  
Ricardo G. Durán ◽  
Claudio Padra ◽  
Rodolfo Rodríguez
Keyword(s):  
Finite Element ◽  
Eigenvalue Problems ◽  
Finite Element Approximation ◽  
Posteriori Error ◽  
Higher Order ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Higher Order Terms

This paper deals with a posteriori error estimators for the linear finite element approximation of second-order elliptic eigenvalue problems in two or three dimensions. First, we give a simple proof of the equivalence, up to higher order terms, between the error and a residual type error estimator. Second, we prove that the volumetric part of the residual is dominated by a constant times the edge or face residuals, again up to higher order terms. This result was not known for eigenvalue problems.

scholarly journals A priori and a posteriori error analysis for a hybrid formulation of a prestressed shell model.

2021 ◽  
Author(s):  
Serge Nicaise ◽  
Ismail Merabet ◽  
Rayhana REZZAG BARA
Keyword(s):  
Shell Model ◽  
A Priori ◽  
Functional Space ◽  
Finite Element Approximation ◽  
Posteriori Error ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
A Priori Error Estimate

This work deals with the finite element approximation of a prestressed shell model using a new formulation where the unknowns (the displacement and the rotation of fibers normal to the midsurface) are described in Cartesian and local covariant basis respectively. Due to the constraint involved in the definition of the functional space, a penalized version is then considered. We obtain a non robust a priori error estimate of this penalized formulation, but a robust one is obtained for its mixed formulation. Moreover, we present a reliable and efficient a posteriori error estimator of the penalized formulation. Numerical tests are included that confirmthe efficiency of our residual a posteriori estimator.

scholarly journals Robust a posteriori error estimation for mixed finite element approximation of linear poroelasticity

2020 ◽  
Author(s):  
Arbaz Khan ◽  
David J Silvester
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Finite Element Approximation ◽  
Posteriori Error ◽  
Physical Parameters ◽  
Element Approximation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Solution Algorithms ◽  
Consolidation Model

Abstract This work is dedicated to the memory of John W. Barrett, who introduced the concept of inf–sup stability to the corresponding author in the bar at the MAFELAP conference in 1981. We analyze a posteriori error estimators for locking-free mixed finite element approximation of Biot’s consolidation model. Three estimators are described. The simplest of these is a conventional residual-based estimator. We establish bounds relating the estimated and true errors, and show that these are independent of the physical parameters. The other two estimators require the solution of local problems. These local problem estimators are also shown to be reliable, efficient and robust. Numerical results are presented that validate the theoretical estimates, and illustrate the effectiveness of the estimators in guiding adaptive solution algorithms. The IFISS and T-IFISS software packages used for the computational experiments are available online.

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