A new 3D equilibrated residual method improving accuracy and efficiency of flux-free error estimates

Author (s): Parés, N.; and Díez, P.
Journal: International Journal for Numerical Methods in Engineering

Volume: 120
Pages: 391 – 432
Date: 2019

Abstract:
The paper presents a novel strategy providing fully computable upper bounds for the energy norm of the error in the context of three-dimensional linear finite element approximations of the reaction-diffusion equation. The upper bounds are guaranteed regardless the size of the finite element mesh and the given data, and all the constants involved are fully computable. The upper bounds property holds if the shape of the domain is polyhedral and the Dirichlet boundary conditions are piecewise-linear. The new approach is an extension of the flux-free methodology introduced by Parés and Díez [1], which introduces a guaranteed, low-cost and efficient flux-free method substantially reducing the computational cost of obtaining guaranteed bounds using flux-free methods while retaining the good quality of the bounds. Besides extending the 2D methodology, specific new modifications are introduced to further reduce the computational cost in the three-dimensional setting. The presented methodology also provides a new strategy to obtain equilibrated boundary tractions which improves the quality of standard techniques while having a similar computational cost.

  
  

Bibtex:

@article{NME:PD2019,
author = {Par\'es, N\'uria and D\i'ez, Pedro},
title = {A new 3D equilibrated residual method improving accuracy and efficiency of flux-free error estimates},
journal = {International Journal for Numerical Methods in Engineering},
volume = {0},
number = {ja},
pages = {},
keywords = {Verification, Adaptivity, Error estimation, Exact/guaranteed/strict bounds, Flux-free, Equilibrated boundary tractions},
doi = {10.1002/nme.6141},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.6141},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.6141},
abstract = {Summary The paper presents a novel strategy providing fully computable upper bounds for the energy norm of the error in the context of three-dimensional linear finite element approximations of the reaction-diffusion equation. The upper bounds are guaranteed regardless the size of the finite element mesh and the given data, and all the constants involved are fully computable. The upper bounds property holds if the shape of the domain is polyhedral and the Dirichlet boundary conditions are piecewise-linear. The new approach is an extension of the flux-free methodology introduced by Par\'es and D\'iez[1], which introduces a guaranteed, low-cost and efficient flux-free method substantially reducing the computational cost of obtaining guaranteed bounds using flux-free methods while retaining the good quality of the bounds. Besides extending the 2D methodology, specific new modifications are introduced to further reduce the computational cost in the three-dimensional setting. The presented methodology also provides a new strategy to obtain equilibrated boundary tractions which improves the quality of standard techniques while having a similar computational cost.},
}