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

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 flux-free 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.

[1]  Ricardo H. Nochetto,et al.  Local problems on stars: A posteriori error estimators, convergence, and performance , 2003, Math. Comput..

[2]  Sergey Korotov,et al.  Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions , 2007 .

[3]  Pedro Díez,et al.  Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates , 2003 .

[4]  Pierre Ladevèze,et al.  Error Estimate Procedure in the Finite Element Method and Applications , 1983 .

[5]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[6]  W. G. Szymczak,et al.  Adaptivity and Error Estimation for the Finite Element Method Applied to Convection Diffusion Problems , 1984 .

[7]  Pedro Díez,et al.  Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the discontinuous Galerkin time discretization , 2008 .

[8]  J. Oden,et al.  Goal-oriented error estimation and adaptivity for the finite element method , 2001 .

[9]  Anthony T. Patera,et al.  A hierarchical duality approach to bounds for the outputs of partial differential equations , 1998 .

[10]  Carsten Carstensen,et al.  Fully Reliable Localized Error Control in the FEM , 1999, SIAM J. Sci. Comput..

[11]  Dietrich Braess,et al.  Equilibrated residual error estimator for edge elements , 2007, Math. Comput..

[12]  M. Křížek.  Conforming equilibrium finite element methods for some elliptic plane problems , 1983 .

[13]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[14]  Pedro Díez,et al.  Subdomain-based flux-free a posteriori error estimators , 2006 .

[15]  J. Peraire,et al.  A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations , 1997 .

[16]  Antonio Huerta,et al.  Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation , 2004, SIAM J. Numer. Anal..

[17]  J. Peraire,et al.  Computing upper and lower bounds for the J-integral in two-dimensional linear elasticity , 2006 .

[18]  Sergey Repin,et al.  Functional a posteriori estimates for the reaction–diffusion problem , 2006 .

[19]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[20]  M. Giles,et al.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002, Acta Numerica.

[21]  Barbara I. Wohlmuth,et al.  A Local A Posteriori Error Estimator Based on Equilibrated Fluxes , 2004, SIAM J. Numer. Anal..

[22]  Rüdiger Verfürth.  Robust A Posteriori Error Estimates for Nonstationary Convection-Diffusion Equations , 2005, SIAM J. Numer. Anal..

[23]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[24]  Anthony T. Patera,et al.  A flux-free nodal Neumann subproblem approach to output bounds for partial differential equations , 2000 .

[25]  Antonio Huerta,et al.  The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations , 2006 .

[26]  Ivo Babuška,et al.  A posteriori error estimation for generalized finite element methods , 2006 .

[27]  Martin Vohralík.  A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization , 2008 .

[28]  Núria Parés Mariné.  Error assessment for functional outputs of pde's: bounds and goal-oriented adaptivity , 2005 .

[29]  J. T. Oden,et al.  A posteriori error estimation of finite element approximations in fluid mechanics , 1990 .

[30]  Philippe Destuynder,et al.  Explicit error bounds in a conforming finite element method , 1999, Math. Comput..

[31]  A. Huerta,et al.  Bounds of functional outputs for parabolic problems. Part II: Bounds of the exact solution , 2008 .

[32]  Jaime Peraire,et al.  Computing Bounds for Linear Functionals of Exact Weak Solutions to the Advection-Diffusion-Reaction Equation , 2005, SIAM J. Sci. Comput..

[33]  Rüdiger Verfürth,et al.  Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation , 1998 .

[34]  Ivo Babuška,et al.  Reliable and Robust A Posteriori Error Estimation for Singularly Perturbed Reaction-Diffusion Problems , 1999 .

[35]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[36]  J. Tinsley Oden,et al.  Computable Error Estimators and Adaptive Techniques for Fluid Flow Problems , 2003 .

[37]  W. Wendland,et al.  A‐posteriori error analysis and adaptive finite element methods for singularly perturbed convection‐diffusion equations , 1982 .

[38]  Pedro Díez,et al.  Guaranteed energy error bounds for the Poisson equation using a flux‐free approach: Solving the local problems in subdomains , 2009 .

[39]  Rüdiger Verfürth,et al.  Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations , 2005, SIAM J. Numer. Anal..

[40]  Tomáš Vejchodský,et al.  Guaranteed and locally computable a posteriori error estimate , 2006 .