2021/10/05 by Ludovic Chamoin, Chamoin, Ludovic, Frédéric Legoll +1
Decision Sciences · Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2110.02160
openalex publication_date 2021/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article is a review on basic concepts and tools devoted to a posteriori\nerror estimation for problems solved with the Finite Element Method. For the\nsake of simplicity and clarity, we mostly focus on linear elliptic diffusion\nproblems, approximated by a conforming numerical discretization. The review\nmainly aims at presenting in a unified manner a large set of powerful\nverification methods, around the concept of equilibrium. Methods based on that\nconcept provide error bounds that are fully computable and mathematically\ncertified. We discuss recovery methods, residual methods, and duality-based\nmethods for the estimation of the whole solution error (i.e. the error in\nenergy norm), as well as goal-oriented error estimation (to assess the error on\nspecific quantities of interest). We briefly survey the possible extensions to\nnon-conforming numerical methods, as well as more complex (e.g. nonlinear or\ntime-dependent) problems. We also provide some illustrating numerical examples\non a linear elasticity problem in 3D.\n