Error Estimation and Quality Control

Self Cite

SUMMARYThe paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The bounds are computed using implicit a-posteriori error estimators from stabilized finite element approximations of the exact solution. A new methodology is introduced, based in the ideas presented in [1] for the Galerkin formulation, that allows obtaining bounds also for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the fluxfree technique.

“…Moreover, for a fixed q ∈ N the optimal choice is to determineq andr verifying (15) and minimizing the upper bound…”

Section: Theoremmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Goal Oriented Simulations: Outputs and Adjoint Problemmentioning

confidence: 99%

“…(Ω) be two dual estimates fulfilling equation (15). Then, an upper bound for the energy norm of the solution z of (14) is computed as…”

Section: Theoremmentioning

confidence: 99%

See 3 more Smart Citations

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Mariné

Dı́ez

Huerta

2012

Self Cite

Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales

Paladim

Almeida

Bordas

et al. 2016

This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion coefficients. More precisely, in the context of linear elliptic diffusion problems in randomly packed particulate composites, we develop an approach to strictly bound the error in the expectation and second moment of quantities of interest, without ever solving the fine‐scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine‐scale features. Our computations are purely macroscopic, deterministic and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager–Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss–Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation. Copyright © 2016 John Wiley & Sons, Ltd.

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

Mariné

Dı́ez

2019

Self Cite

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 bound 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 in the paper “A new equilibrated residual method improving accuracy and efficiency of flux‐free error estimates”, 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.