Convergence of Adaptive Finite Element Methods

By Pedro Morin, Ricardo H. Nochetto, Kunibert G. Siebert. SIAM Review, 2002

401 citations recorded by the source corpus.

Adaptive finite element methods (FEMs) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Subfields: Computational Mechanics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/7022a9e2c64eff3fc83ba1f50e290aaf6a33b013

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Electrical and Electronic Engineering

Citations in the corpus

Primer of Adaptive Finite Element Methods (2011, methodology): “We refer to [7, 9, 16, 17, 23, 24, 37, 40–42] for other approaches and to Sect.”
Theory of adaptive finite element methods: An introduction (2009, methodology): “R. H. Nochetto, K. G. Siebert, and A. Veeser”
Guaranteed and locally computable a posteriori error estimate (2006, methodology): “Let us note that the proposed error estimator can be employed also in more elaborate adaptive strategies, such as the convergent strategy described in Morin et al. (2002).”
Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM (2009, methodology): “Quasi-optimal rates of convergence for elliptic problems with smooth coefficients on polygonal domains were obtained before in [4,11,13,26,47,56] using adaptive meshes (usually graded meshes, like in this paper).”
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].”
Adaptive finite element heterogeneous multiscale method for homogenization problems (2011, methodology): “2 of [33], we use here a two-dimensional, highly oscillating conductivity tensor.”
Quasioptimal cardinality of AFEM driven by nonresidual estimators (2012, background): “It is thus intriguing why non-residual estimators yield a prctical performance of AFEM similar to residual-type estimators (see Binevet al., 2004; Cascónet al., 2008; Mekchay & Nochetto, 2005; Morin et al., 2000, 2002; Stevenson, 2007).”
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method (2008, neutral): “, |||u− Uk|||Ω = |||u− Uk+1|||Ω; see [17, 18] for further details.”
ANALYSIS OF THE FINITE ELEMENT METHOD FOR TRANSMISSION/MIXED BOUNDARY VALUE PROBLEMS ON GENERAL POLYGONAL DOMAINS ∗ (2010, background): “in the study of the adaptive Finite Element Method, is a promising future direction of our research [56, 60].”