Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation

2009

SIAM J. Sci. Comput.

Abstract. The 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 proposed approach is an alternative to the standard residual type estimators (hybrid-flux), circumventing the need of flux-equilibration following a fluxfree error estimation strategy. The presented estimator provides sharper estimates than the ones provided by both the standard hybrid-flux techniques and other flux-free techniques.

Section: Upper Bounds For the Energy Norm: Complementary Energy Relaxmentioning

confidence: 99%

“…2) may be found in [29,30]. Note that if one choosesr = 0, the existence of a piecewise polynomial dual estimateq verifying…”

Section: Remark 3 Note That the Relaxed Problem (42) Admits At Leasmentioning

confidence: 99%

See 1 more Smart Citation

Exact Bounds for Linear Outputs of the Advection-Diffusion-Reaction Equation Using Flux-Free Error Estimates

Mariné¹,

Dı́ez

2009

SIAM J. Sci. Comput.

“…Since the mid 2000s, attention has been devoted to provide certified bounds for quantities of interest [2,3,4,5,6,7,8,9]. In particular [1] presents a comparison of the performance of two of the main techniques to compute guaranteed bounds for quantities of interest in the context of the advection-reaction-diffusion equation: a standard residual type estimator (hybrid-flux) proposed in [10] and the new flux-free technique proposed in [11].…”

Section: Introductionmentioning

confidence: 99%

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

Mariné

Dı́ez

Numerical Meth Engineering

2012

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.

“…Bounds for the exact solution of the boundary value problem as presented in [3,26,22] are not addressed here.…”

Section: Introductionmentioning

confidence: 99%

Goal Oriented Error Estimation for the Element Free Galerkin Method

Vidal

Lecture Notes in Computational Science and Engineering

Summary.A novel approach for implicit residual-type error estimation in meshfree methods is presented. This allows to compute upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods.