An optimal control approach to a posteriori error estimation in finite element methods

By Roland Becker, Rolf Rannacher. Acta Numerica, 2001

1,339 citations recorded by the source corpus.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/34a6f4386e8967efa7395ee591abf9a2da548c9a

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In Computational Mechanics

In Numerical Analysis

Citations in the corpus

Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs (2012, background): “Adjoint models are key ingredients in many algorithms of computational science, such as parameter identification [40], sensitivity analysis [6], data assimilation [28], optimal control [30], adaptive observations [42], predictability analysis [36], and error estimation [4].”
Anisotropic mesh refinement for discontinuous Galerkin methods in two‐dimensional aerodynamic flow simulations (2008, methodology): “[16, 17] and the references cited therein, used to estimate the error in specific target quantities evaluated based on the numerical solution which may serve as a stopping criterion of an adaptive refinement algorithm when a”
Adaptive reduced basis finite element heterogeneous multiscale method (2013, background): “While there is a large literature for single scale adaptive methods (see for example [15, 16, 17, 18] and the references therein), adaptive multiscale methods are still at an early stage of development.”
Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics (2011, methodology): “Becker and Rannacher [14] show that the error estimate can be improved to third order by including the residual of the adjoint problem and R 3 ku h k; k h k is a remainder that is third order in the primal and adjoint error.”
Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows (2003, methodology): “Becker, Rannacher and collaborators [8,9,11,42] have developed an optimal control approach for output-based grid adaptation within a Galerkin finite element framework.”
Finite Element Methods (2000, methodology): “The justification of some substitute |D2hzh|L2(T ) (postprocessed with some averaging technique) for |z|H2(T ) is through striking numerical evidence; we refer to Becker and Rannacher (2001) and Bangerth and Rannacher (2003) for details and numerical experiments.”
A review of numerical methods for nonlinear partial differential equations (2012, background): “The finite elements are adapted to local variations of the solution: adaptivity is often tuned by a posteriori estimates [14, 22, 157].”
Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (2012, methodology): “(1995); Becker and Rannacher (2001). This technique is based on the solution of an auxiliary linearized dual (adjoint) problem in order to estimate the error in a given goal functional.”
Optimal Control and Numerical Adaptivity for Advection-Diffusion Equations (2005, methodology): “• the error indicator [3]: η E := ∑ K∈Th hK { (ρK) 2 + (ρpK) 2 + (ρK) 2 } 1 2 , (63)”
SOLUTIONS OF 3D NAVIER-STOKES BENCHMARK PROBLEMS WITH ADAPTIVE FINITE ELEMENTS (2006, methodology): “Here, we applied the dual-weighted residual method of Becker & Rannacher [7].”