scholarly journals An Adaptive Finite Element Method for Stationary Incompressible Thermal Flow Based on Projection Error Estimation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Zhili Guo ◽  
Jian Su ◽  
Hao Chen ◽  
Xiaomin Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stationary Flow ◽  
Adaptive Strategy ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Thermal Flow ◽  
Element Method

An adaptive finite element method is presented for the stationary incompressible thermal flow problems. A reliable a posteriori error estimator based on a projection operator is proposed and it can be computed easily and implemented in parallel. Finally, three numerical examples are given to illustrate the efficiency of the adaptive finite element method. We also show that the adaptive strategy is effective to detect local singularities in the physical model of square cavity stationary flow in the third example.

3D direct current resistivity modeling with unstructured mesh by adaptive finite-element method

Geophysics ◽  
2010 ◽  
Vol 75 (1) ◽  
pp. H7-H17 ◽  
Author(s):  
Zhengyong Ren ◽  
Jingtian Tang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Direct Current ◽  
Homogeneous Model ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Resistivity Modeling ◽  
Element Method

A new adaptive finite-element method for solving 3D direct-current resistivity modeling problems is presented. The method begins with an initial coarse mesh, which is then adaptively refined wherever a gradient-recovery-based a posteriori error estimator indicates that refinement is necessary. Then the problem is solved again on the new grid. The alternating solution and refinement steps continue until a given error criterion is satisfied. The method is demonstrated on two synthetic resistivity models with known analytical solutions, so the errors can be quantified. The applicability of the numerical method is illustrated on a 2D homogeneous model with a topographic valley. Numerical results show that this method is efficient and accurate for geometrically complex situations.

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.

scholarly journals Quasi-optimality of an Adaptive Finite Element Method for Cathodic Protection

2019 ◽  
Vol 53 (5) ◽  
pp. 1645-1665
Author(s):  
Guanglian Li ◽  
Yifeng Xu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cathodic Protection ◽  
Finite Element Approximation ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Energy Error ◽  
Element Method

In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2D cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We propose a standard adaptive finite element method involving the Dörfler marking and a minimal refinement without the interior node property. Furthermore, we establish the contraction property of this adaptive algorithm in terms of the sum of the energy error and the scaled estimator. This essentially allows for a quasi-optimal convergence rate in terms of the number of elements over the underlying triangulation. Numerical experiments are provided to confirm this quasi-optimality.

Convergent Adaptive Finite Element Method Based on Centroidal Voronoi Tessellations and Superconvergence

2011 ◽  
Vol 10 (2) ◽  
pp. 339-370 ◽  
Author(s):  
Yunqing Huang ◽  
Hengfeng Qin ◽  
Desheng Wang ◽  
Qiang Du
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Voronoi Tessellation ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Adaptive Procedure ◽  
Seamless Integration ◽  
Practical Applications ◽  
Element Method

AbstractWe present a novel adaptive finite element method (AFEM) for elliptic equations which is based upon the Centroidal Voronoi Tessellation (CVT) and superconvergent gradient recovery. The constructions of CVT and its dual Centroidal Voronoi Delaunay Triangulation (CVDT) are facilitated by a localized Lloyd iteration to produce almost equilateral two dimensional meshes. Working with finite element solutions on such high quality triangulations, superconvergent recovery methods become particularly effective so that asymptotically exact a posteriori error estimations can be obtained. Through a seamless integration of these techniques, a convergent adaptive procedure is developed. As demonstrated by the numerical examples, the new AFEM is capable of solving a variety of model problems and has great potential in practical applications.

An Optimal Adaptive Finite Element Method for an Obstacle Problem

2015 ◽  
Vol 15 (3) ◽  
pp. 259-277 ◽  
Author(s):  
Carsten Carstensen ◽  
Jun Hu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Obstacle Problem ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
A Posteriori ◽  
Optimal Complexity ◽  
A Posteriori Error ◽  
Element Method

AbstractThis paper provides a refined a posteriori error control for the obstacle problem with an affine obstacle which allows for a proof of optimal complexity of an adaptive algorithm. This is the first adaptive mesh-refining finite element method known to be of optimal complexity for some variational inequality. The result holds for first-order conforming finite element methods in any spacial dimension based on shape-regular triangulation into simplices for an affine obstacle. The key contribution is the discrete reliability of the a posteriori error estimator from [Numer. Math. 107 (2007), 455–471] in an edge-oriented modification which circumvents the difficulties caused by the non-existence of a positive second-order approximation [Math. Comp. 71 (2002), 1405–1419].

Adaptive finite element method for thermal flow problems

AIAA Journal ◽  
1994 ◽  
Vol 32 (4) ◽  
pp. 741-747 ◽  
Author(s):  
Dominique Pelletier ◽  
Jean-Francois Hetu ◽  
Florin Ilinca
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Thermal Flow ◽  
Flow Problems ◽  
Element Method

scholarly journals An adaptive finite element method for Fredholm integral equations of the first kind and its verification on experimental data

2013 ◽  
Vol 11 (8) ◽  
Author(s):  
Nikolay Koshev ◽  
Larisa Beilina
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Bounded Operator ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Fredholm Integral Equations ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
Element Method

AbstractWe propose an adaptive finite element method for the solution of a linear Fredholm integral equation of the first kind. We derive a posteriori error estimates in the functional to be minimized and in the regularized solution to this functional, and formulate corresponding adaptive algorithms. To do this we specify nonlinear results obtained earlier for the case of a linear bounded operator. Numerical experiments justify the efficiency of our a posteriori estimates applied both to the computationally simulated and experimental backscattered data measured in microtomography.

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