The finite element method and its reliability

By Ivo Babuska, Theofanis Strouboulis. 2001

847 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/d1e00887d2be9ebf1e3f070cb6c2c42664b75cc9

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Citations in the corpus

A New Finite Element Gradient Recovery Method: Superconvergence Property (2005, background): “In the literature, we know that the tangential derivative of the finite element solution is superconvergent at two Gaussian points along the element edge for certain uniform mesh patterns [2] [3] [17].”
A posteriori error estimation and dynamic adaptivity for symmetric discontinuous Galerkin approximations of reactive transport problems (2006, methodology): “A posteriori error estimators can be used to signify where modifications in discretization parameters need to be made, thus achieving adaptivity, in particular, goal-oriented hp-adaptivity [31,39,20,8,5–7,19,21–23,26,32,35,38,40,44,27,18,4,2,3], see also [57,9,13] and references therein.”
The p-Version of the Finite Element and Finite Cell Methods (2017, background): “The definitive properties of bilinear forms and linear functionals are listed, for example, in Szabó and Babuška (1991), Schwab (1998), and Babuška and Strouboulis (2001). The definitions for (u, v),  (v), X, and Y depend on the choice of the generalized formulation and the boundary conditions.”
Functional a posteriori estimates for the reaction–diffusion problem (2006, background): “We omit here a detailed review of the literature but refer to the monographs [1,4,8] and the references therein.”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, methodology): “Some sources of errors arising in computer simulations now can be controlled and reduced by using sophisticated techniques such as a posteriori error estimation [1, 3, 57], mesh adaptivity, and the more recent modeling error analysis [49, 50, 13].”
Finite Element Methods (2000, methodology): “Details on the implementation are given in Ainsworth and Oden (2000); a detailed error analysis with higher-order equilibrations and the perturbation by a finite element simulation of the local problems with corrections can be found in Ainsworth and Oden (2000) and Babuška and Strouboulis (2001).”
AN A POSTERIORI ERROR ANALYSIS OF ADAPTIVE FINITE ELEMENT METHODS FOR DISTRIBUTED ELLIPTIC CONTROL PROBLEMS WITH CONTROL CONSTRAINTS (2008, background): “, the monographs [1,3,4,14,26,27] and the references therein).”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, background): “There is, thus, even greater need for a posteriori error estimation in the RB context than in the much more studied FE context [4, 5, 12, 11, 13, 22].”
Discontinuous Galerkin methods for reactive transport in porous media (2003, methodology): “A posteriori error estimators can be used to signify where modifications in discretization parameters need to be made and thus to achieve adaptivity, in particular the goal-oriented hp-adaptivity [1– 4, 21, 28–30, 32–34, 51, 58, 61, 64, 67–69, 86], see also [5, 12, 96] and references therein.”
Some remarks on the history and future of averaging techniques in a posteriori finite element error analysis (2004, methodology): “Details on the implementation can be found in [1], a related correction technique in [6].”
Estimation of interpolation error constants for the P0 and P1 triangular finite elements (2007, methodology): “Thus such an evaluation has become progressively more important and has been attempted especially for adaptive finite element calculations based on a posteriori error estimation as well as for numerical verification by FEM [1,4,6,7,13].”
THE POLYNOMIAL-PRESERVING RECOVERY FOR HIGHER ORDER FINITE ELEMENT METHODS IN 2D AND 3D (2005, background): “For the history and the advances in this field, the reader is referred to [1, 5, 8, 15, 23].”