Variational Multiscale A Posteriori Error Estimation for Quantities of Interest

2009 ◽  
Vol 76 (2) ◽  
Author(s):  
Guillermo Hauke ◽  
Daniel Fuster
Keyword(s):  
Error Estimates ◽  
Posteriori Error ◽  
Local Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Variational Multiscale ◽  
Stabilized Methods ◽  
Global And Local ◽  
Quantities Of Interest

This paper applies the variational multiscale theory to develop an explicit a posteriori error estimator for quantities of interest and linear functionals of the solution. The method is an extension of a previous work on global and local error estimates for solutions computed with stabilized methods. The technique is based on approximating an exact representation of the error formulated as a function of the fine-scale Green function. Numerical examples for the multidimensional transport equation confirm that the method can provide good local error estimates of quantities of interest both in the diffusive and the advective limit.

scholarly journals Multigoal-oriented error estimates for non-linear problems

2019 ◽  
Vol 27 (4) ◽  
pp. 215-236 ◽  
Author(s):  
Bernhard Endtmayer ◽  
Ulrich Langer ◽  
Thomas Wick
Keyword(s):  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Nonlinear Problems ◽  
Posteriori Error ◽  
Stopping Rules ◽  
Nonlinear Pdes ◽  
Error Estimator ◽  
Mesh Adaptivity ◽  
A Posteriori ◽  
A Posteriori Error

Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

scholarly journals On a multiscale a posteriori error estimator for the stokes and Brinkman equations

2019 ◽  
Author(s):  
Rodolfo Araya ◽  
Ramiro Rebolledo ◽  
Frédéric Valentin
Keyword(s):  
Level Structure ◽  
Adaptive Strategy ◽  
Posteriori Error ◽  
Local Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Brinkman Equations

Abstract This work proposes and analyzes a residual a posteriori error estimator for the multiscale hybrid-mixed (MHM) method for the Stokes and Brinkman equations. The error estimator relies on the multi-level structure of the MHM method and considers two levels of approximation of the method. As a result the error estimator accounts for a first-level global estimator defined on the skeleton of the partition and second-level contributions from element-wise approximations. The analysis establishes local efficiency and reliability of the complete multiscale estimator. Also, it yields a new face-adaptive strategy on the mesh’s skeleton, which avoids changing the topology of the global mesh. Specially designed to work on multiscale problems, the present estimator can leverage parallel computers since local error estimators are independent of each other. Academic and realistic multiscale numerical tests assess the performance of the estimator and validate the adaptive algorithms.

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