scholarly journals A PRIORI ERROR ESTIMATES FOR ENERGY-BASED QUASICONTINUUM APPROXIMATIONS OF A PERIODIC CHAIN

2011 ◽  
Vol 21 (12) ◽  
pp. 2491-2521 ◽  
Author(s):  
CHRISTOPH ORTNER ◽  
HAO WANG
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Stability Estimates ◽  
Norm Estimates ◽  
Priori Error Estimates ◽  
Sharp Stability ◽  
Consistency Error ◽  
Periodic Chain ◽  
Homogeneous Strain

We derive a priori error estimates for three prototypical energy-based quasicontinuum (QC) methods: the local QC method, the energy-based QC method, and the quasi-nonlocal QC method. Our analysis decomposes the consistency error into modeling and coarsening errors. While previous results on estimating the modeling error exist, we present a new and simpler proof based on negative-norm estimates. Our stability analysis extends previous results on sharp stability estimates under homogeneous strain to the nonlinear setting. Finally, we present numerical experiments to illustrate the results of our analysis.

scholarly journals A note on a priori error estimates for augmented mixed methods

2016 ◽  
Vol 57 ◽  
pp. 139-144
Author(s):  
Tomás P. Barrios ◽  
Edwin Behrens ◽  
Rommel Bustinza
Keyword(s):  
Mixed Methods ◽  
Error Estimates ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Priori Error Estimates

scholarly journals A priori error estimates for a linearized fracture control problem

2020 ◽  
Author(s):  
Masoumeh Mohammadi ◽  
Winnifried Wollner
Keyword(s):  
Control Problem ◽  
Error Estimates ◽  
Optimization Problems ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Adjoint Variables ◽  
Conforming Finite Element Method ◽  
Priori Error Estimates ◽  
Fracture Control ◽  
Control State

Abstract A control problem for a linearized time-discrete regularized fracture propagation process is considered. The discretization of the problem is done using a conforming finite element method. In contrast to many works on discretization of PDE constrained optimization problems, the particular setting has to cope with the fact that the linearized fracture equation is not necessarily coercive. A quasi-best approximation result will be shown in the case of an invertible, though not necessarily coercive, linearized fracture equation. Based on this a priori error estimates for the control, state, and adjoint variables will be derived.

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