AN ADAPTIVE FINITE ELEMENT METHOD WITH EFFICIENT MAXIMUM NORM ERROR CONTROL FOR ELLIPTIC PROBLEMS

1994 ◽  
Vol 04 (03) ◽  
pp. 313-329 ◽  
Author(s):  
KENNETH ERIKSSON
Keyword(s):  
Finite Element ◽  
Error Control ◽  
A Priori ◽  
Mesh Size ◽  
Maximum Norm ◽  
Adaptive Finite Element Method ◽  
Local Regularity ◽  
Adaptive Finite Element ◽  
Element Discretization ◽  
Elliptic Model

A posteriori and a priori error estimates are derived for a finite element discretization method applied to an elliptic model problem. The underlying partitions need not be quasi-uniform and can be highly graded; only a certain weak, local mesh regularity is assumed. The error is bounded in terms of the local mesh size and the local regularity of the solution and data. An adaptive algorithm is designed for automatic control of the discretization error in the maximum norm. The error control is proved to be both reliable and efficient.

scholarly journals Adaptive modelling of variably saturated seepage problems

2021 ◽  
Author(s):  
B Ashby ◽  
C Bortolozo ◽  
A Lukyanov ◽  
T Pryer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Water Flux ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
Element Method

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

CONVERGENCE OF A hp-STREAMLINE DIFFUSION SCHEME FOR VLASOV–FOKKER–PLANCK SYSTEM

2007 ◽  
Vol 17 (08) ◽  
pp. 1159-1182 ◽  
Author(s):  
M. ASADZADEH ◽  
A. SOPASAKIS
Keyword(s):  
Finite Element ◽  
A Priori ◽  
Mesh Size ◽  
Stability Estimates ◽  
Spectral Order ◽  
Element Discretization ◽  
Diffusion Scheme ◽  
Streamline Diffusion ◽  
Stabilization Parameter ◽  
Fokker Planck

We analyze the hp-version of the streamline diffusion finite element method for the Vlasov–Fokker–Planck system. For this method we prove the stability estimates and derive sharp a priori error bounds in a stabilization parameter δ ~ min (h/p, h2/σ), with h denoting the mesh size of the finite element discretization in phase-space-time, p the spectral order of approximation, and σ the transport cross-section.

scholarly journals Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

2012 ◽  
Vol 11 (1) ◽  
pp. 179-214 ◽  
Author(s):  
M. Holst ◽  
J.A. McCammon ◽  
Z. Yu ◽  
Y.C. Zhou ◽  
Y. Zhu
Keyword(s):  
Finite Element ◽  
Boltzmann Equation ◽  
Geometric Model ◽  
A Priori ◽  
Molecular Surface ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Regularization Technique ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation

AbstractWe consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a prioriL∞ estimates. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

Adaptive Finite Element Methods: A Review

1997 ◽  
Vol 50 (10) ◽  
pp. 581-591 ◽  
Author(s):  
Long-yuan Li ◽  
Peter Bettess
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Element Model ◽  
Adaptive Finite Element Method ◽  
Finite Element Models ◽  
Adaptive Finite Element ◽  
Element Discretization ◽  
New Model ◽  
Discretization Errors ◽  
The Finite Element Model

The adaptive finite element method (FEM) was developed in the early 1980s. The basic concept of adaptivity developed in the FEM is that, when a physical problem is analyzed using finite elements, there exist some discretization errors caused owing to the use of the finite element model. These errors are calculated in order to assess the accuracy of the solution obtained. If the errors are large, then the finite element model is refined through reducing the size of elements or increasing the order of interpolation functions. The new model is re-analyzed and the errors in the new model are recalculated. This procedure is continued until the calculated errors fall below the specified permissible values. The key features in the adaptive FEM are the estimation of discretization errors and the refinement of finite element models. This paper presents a brief review of the methods for error estimates and adaptive refinement processes applied to finite element calculations. The basic theories and principles of estimating finite element discretization errors and refining finite element models are presented. This review article contains 131 references.

Three Dimensional Dynamic Analyses on Stroller Wheel with Shock Absorber

2013 ◽  
Vol 387 ◽  
pp. 159-163
Author(s):  
Yi Chern Hsieh ◽  
Minh Hai Doan ◽  
Chen Tai Chang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Three Dimensional ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
The Road ◽  
Dimensional Finite Element ◽  
On The Road ◽  
Three Dimensional Finite Element ◽  
Element Method

We present the analyses of dynamics behaviors on a stroller wheel by three dimensional finite element method. The vibration of the wheel system causes by two different type barriers on the road as an experiment design to mimic the real road conditions. In addition to experiment analysis, we use two different packages to numerically simulate the wheel system dynamics activities. Some of the simulation results have good agreement with the experimental data in this research. Other interesting data will be measured and analyzed by us for future study and we will investigate them by using adaptive finite element method for increasing the precision of the computation results.

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