A flux-free a posteriori error estimator for the incompressible Stokes problem using a mixed FE formulation |
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| Authors: | Fredrik Larsson Pedro Díez Antonio Huerta |
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| Affiliation: | 1. Department of Applied Mechanics, Chalmers University of Technology, SE41296 Göteborg, Sweden;2. Laboratori de Càlcul Numèric, Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Spain;1. Universitat Politècnica de Catalunya, Jordi Girona 1-3, Edifici C1, E-08034 Barcelona, Spain;2. Centre Internacional de Mètodes Numèrics en Enginyeria, Parc Mediterrani de la Tecnologia, Esteve Terrades 5, E-08860 Castelldefels, Spain;1. Collaborative Innovation Centre of Mathematics, School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;2. Department of Applied Mathematics, Chung Yuan Christian University, Jhongli City, Taoyuan County 32023, Taiwan;3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore;4. Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan;1. Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel;2. Afeka, Tel Aviv Academic College of Engineering, Israel;1. Department of Civil and Environmental Engineering, University of Tennessee, Knoxville, 318 John D. Tickle Engineering Building, Knoxville, TN 37996-2313, United States;2. Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 3129E Newmark Civil Engineering Laboratory, MC-250, Urbana, IL 61801-2352, United States;1. Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia;2. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong Special Administrative Region;3. Department of Mathematics, Texas A&M University, College Station, TX, USA;4. Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX, USA |
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| Abstract: | In this contribution, we present an a posteriori error estimator for the incompressible Stokes problem valid for a conventional mixed FE formulation. Due to the saddle-point property of the problem, conventional error estimators developed for pure minimization problems cannot be utilized straight-forwardly. The new estimator is built up by two key ingredients. At first, a computed error approximation, exactly fulfilling the continuity equation for the error, is obtained via local Dirichlet problems. Secondly, we adopt the approach of solving local equilibrated flux-free problems in order to bound the remaining, incompressible, error. In this manner, guaranteed upper and lower bounds, of the velocity “energy norm” of the error as well as goal-oriented (linear) output functionals, with respect to a reference (overkill) mesh are obtained. In particular, it should be noted that this approach requires no computation of hybrid fluxes. Furthermore, the estimator is applicable to mixed FE formulations using continuous pressure approximations, such as the Mini and Taylor–Hood class of elements. In conclusion, a few simple numerical examples are presented, illustrating the accuracy of the error bounds. |
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