Adaptive Finite Element Methods with convergence rates

By Peter Binev, Wolfgang Dahmen, Ronald A. DeVore. Numerische Mathematik, 2004

634 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/48161c1704240154263f545040c7d14bab219be9

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In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA (2005, methodology): “For a modification of the convergent algorithm in [50], quasi-optimality of the adaptive method was proved in [16] and [67].”
The phase field method for geometric moving interfaces and their numerical approximations (2019, neutral): “The first technique gives rise so-called residual-based error bounds [181, 328, 60, 338], and the second one is a duality-based technique, called the Dual Weighted Residual method, which leads to goal-oriented error estimates [30].”
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs (2011, methodology): “We should point out that a similar program was developed when solving a single PDE by either adaptive wavelet methods [7,8,14] or by adaptive finite element methods [4,12,21,26].”
Optimality of a Standard Adaptive Finite Element Method (2007, methodology): “Recently, in [4], Binev, Dahmen and DeVore developed an adaptive finite element method which they showed to be of optimal computational complexity.”
Convergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints (2007, methodology): “AFEMs typically consist of successive loops of the sequence SOLVE → ESTIMATE → MARK → REFINE .”
Quasioptimal cardinality of AFEM driven by nonresidual estimators (2012, methodology): “We refer to Cascónet al. (2008) and (Nochettoet al., 2009, Section 9) for a discussion ofAs and to Binev et al. (2004, 2002); Stevenson (2007) for a connection with Besov classes for the Laplacian.”
Morley Finite Element Method for the Eigenvalues of the Biharmonic Operator (2014, methodology): “Generate T ℓ +1 from T ℓ and M ℓ with newest-vertex bisection [4, 39]. end for Output: Triangulations ( T ℓ ) ℓ and discrete solutions ( λ ℓ,j , u ℓ,j ) j ∈ J ℓ .”
Approximation Classes for Adaptive Methods (2002, methodology): “[ 2 ]. We shall restrict our presentation in this paper to this setting.”
The completion of locally refined simplicial partitions created by bisection (2008, methodology): “In [ BDD04 ], considering conforming partitions into triangles generated by the so-called newest vertex bisection rule starting from some fixed initial conforming triangulation, Binev, Dahmen and DeVore showed that the total number of triangles in the partition at termination of the adaptive finite ”
Some remarks on the history and future of averaging techniques in a posteriori finite element error analysis (2004, background): “Coarsening steps may be involved to prove optimal performance of adaptive mesh-refinement [14].”
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method (2008, neutral): “The discussion reveals that separate marking might be sub-optimal and explains why the AFEM designed by Morin, Nochetto, and Siebert [17, 18] is altered in former optimality proofs [ 3 , 23].”
A residual a posteriori error estimate for the Virtual Element Method (2017, background): “In recent years a posteriori error analysis and optimality investigations of steady-state adaptive discretizations have been widely tackled for several discretization approaches and model equations, obtaining several interesting results.(32,43,36,26) A large effort has been recently spent on unstead”
Theory of adaptive finite element methods: An introduction (2009, methodology): “Classical vs Adaptive Approximation in 1d We start with a simple motivation in 1d for the use of adaptive procedures, due to DeVore [28].”
Approximation of high-dimensional parametric PDEs * (2015, background): “Let us point out that a similar objective can be attained when considering the spatial discretization of a single PDE, by either adaptive wavelet methods [21, 22, 38] or by adaptive finite element methods [70, 7, 86].”
AN A POSTERIORI ERROR ANALYSIS OF ADAPTIVE FINITE ELEMENT METHODS FOR DISTRIBUTED ELLIPTIC CONTROL PROBLEMS WITH CONTROL CONSTRAINTS (2008, methodology): “The adaptive refinement process The refinement of the triangulation Th(Ω) is based on a bulk criterion that has been previously used in the convergence analysis of adaptive finite element for nodal finite element methods [8,13,25] and for nonconforming,”
A review of numerical methods for nonlinear partial differential equations (2012, background): “In other cases, one may need to carefully adjust the measure of local errors using norms which are compatible with adaptive grids [22].”