scholarly journals Optimal convergence behavior of adaptive FEM driven by simple (h−h∕2)-type error estimators

2020 ◽  
Vol 79 (3) ◽  
pp. 623-642
Author(s):  
Christoph Erath ◽  
Gregor Gantner ◽  
Dirk Praetorius
Keyword(s):  
Convergence Behavior ◽  
Error Estimators ◽  
Optimal Convergence ◽  
Adaptive Fem

scholarly journals Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form

2017 ◽  
Vol 86 (307) ◽  
pp. 2213-2237 ◽  
Author(s):  
Daniele Boffi ◽  
Dietmar Gallistl ◽  
Francesca Gardini ◽  
Lucia Gastaldi
Keyword(s):  
Mixed Form ◽  
Optimal Convergence ◽  
Adaptive Fem

scholarly journals Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

2014 ◽  
Vol 14 (4) ◽  
pp. 485-508 ◽  
Author(s):  
Michael Feischl ◽  
Thomas Führer ◽  
Gregor Mitscha-Eibl ◽  
Dirk Praetorius ◽  
Ernst P. Stephan
Keyword(s):  
Singular Integral Equations ◽  
Adaptive Mesh ◽  
Weighting Factor ◽  
Residual Error ◽  
Boundary Element Methods ◽  
Error Estimator ◽  
Error Estimators ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
Adaptive Fem

AbstractWe analyze adaptive mesh-refining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEM-BEM or by the residual error estimator introduced by Birgit Faermann for BEM for weakly-singular integral equations. We prove that in either case the usual adaptive algorithm drives the associated error estimator to zero. Emphasis is put on the fact that the error estimators considered are not even globally equivalent to weighted-residual error estimators for which recently convergence with quasi-optimal algebraic rates has been derived.

scholarly journals Optimal Convergence Rates for Goal-Oriented FEM with Quadratic Goal Functional

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Roland Becker ◽  
Michael Innerberger ◽  
Dirk Praetorius
Keyword(s):  
Dual Problem ◽  
Convergence Rates ◽  
Linear Convergence ◽  
Elliptic Pde ◽  
Optimal Convergence ◽  
Optimal Convergence Rates ◽  
Adaptive Fem ◽  
Numer Anal

AbstractWe consider a linear elliptic PDE and a quadratic goal functional. The goal-oriented adaptive FEM algorithm (GOAFEM) solves the primal as well as a dual problem, where the goal functional is always linearized around the discrete primal solution at hand. We show that the marking strategy proposed in [M. Feischl, D. Praetorius and K. G. van der Zee, An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal.54 (2016), 3, 1423–1448] for a linear goal functional is also optimal for quadratic goal functionals, i.e., GOAFEM leads to linear convergence with optimal convergence rates.

Determination of Base Flipping Free Energy Landscapes from Nonequilibrium Stratification

2019 ◽  
Author(s):  
Xiaohui Wang ◽  
Zhaoxi Sun
Keyword(s):  
Free Energy ◽  
Energy Landscape ◽  
Sampling Technique ◽  
Umbrella Sampling ◽  
Energy Simulation ◽  
Nonlinear Dependence ◽  
Free Energy Landscape ◽  
Convergence Criterion ◽  
Base Flipping ◽  
Convergence Behavior

<p>Correct calculation of the variation of free energy upon base flipping is crucial in understanding the dynamics of DNA systems. The free energy landscape along the flipping pathway gives the thermodynamic stability and the flexibility of base-paired states. Although numerous free energy simulations are performed in the base flipping cases, no theoretically rigorous nonequilibrium techniques are devised and employed to investigate the thermodynamics of base flipping. In the current work, we report a general nonequilibrium stratification scheme for efficient calculation of the free energy landscape of base flipping in DNA duplex. We carefully monitor the convergence behavior of the equilibrium sampling based free energy simulation and the nonequilibrium stratification and determine the empirical length of time blocks required for converged sampling. Comparison between the performances of equilibrium umbrella sampling and nonequilibrium stratification is given. The results show that nonequilibrium free energy simulation is able to give similar accuracy and efficiency compared with the equilibrium enhanced sampling technique in the base flipping cases. We further test a convergence criterion we previously proposed and it comes out that the convergence behavior determined by this criterion agrees with those given by the time-invariant behavior of PMF and the nonlinear dependence of standard deviation on the sample size. </p>

Performance of Error Estimators for Classification

2010 ◽  
Vol 5 (1) ◽  
pp. 53-67 ◽  
Author(s):  
Edward Dougherty ◽  
Chao Sima ◽  
> Hua ◽  
Blaise Hanczar ◽  
Ulisses Braga-Neto
Keyword(s):  
Error Estimators
Sign in / Sign up

Export Citation Format

Share Document