Adaptive Uzawa algorithm for the Stokes equation

2019 ◽  
Vol 53 (6) ◽  
pp. 1841-1870
Author(s):  
Giovanni Di Fratta ◽  
Thomas Führer ◽  
Gregor Gantner ◽  
Dirk Praetorius
Keyword(s):  
Stokes System ◽  
Total Error ◽  
Linear Convergence ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Prior Work ◽  
Uzawa Algorithm ◽  
Interior Node ◽  
The Given ◽  
The Stokes Equation

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively,e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.

scholarly journals DESIGN AND CONVERGENCE OF AFEM IN H(DIV)

2007 ◽  
Vol 17 (11) ◽  
pp. 1849-1881 ◽  
Author(s):  
J. MANUEL CASCON ◽  
RICARDO H. NOCHETTO ◽  
KUNIBERT G. SIEBERT
Keyword(s):  
Mixed Boundary ◽  
Total Error ◽  
Posteriori Error ◽  
Coefficient Matrix ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
A Posteriori Error ◽  
Energy Error ◽  
Jump Discontinuities ◽  
Posteriori Error Analysis

We design an adaptive finite element method (AFEM) for mixed boundary value problems associated with the differential operator A-∇div in H(div, Ω). For A being a variable coefficient matrix with possible jump discontinuities, we provide a complete a posteriori error analysis which applies to both Raviart–Thomas ℝ𝕋n and Brezzi–Douglas–Marini 𝔹𝔻𝕄n elements of any order n in dimensions d = 2, 3. We prove a strict reduction of the total error between consecutive iterates, namely a contraction property for the sum of energy error and oscillation, the latter being solution-dependent. We present numerical experiments for ℝ𝕋n with n = 0, 1 and 𝔹𝔻𝕄1 which document the performance of AFEM and corroborate as well as extend the theory.

scholarly journals Microwave thermometry with potential application in non-invasive monitoring of hyperthermia

2020 ◽  
Vol 28 (5) ◽  
pp. 739-750
Author(s):  
Morteza Ghaderi Aram ◽  
Larisa Beilina ◽  
Hana Dobsicek Trefna
Keyword(s):  
Least Squares Method ◽  
Regularization Parameter ◽  
Region Of Interest ◽  
Optimal Choice ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Non Invasive ◽  
Full Wave ◽  
Reconstruction Quality ◽  
Regularization Techniques

AbstractIntegration of an adaptive finite element method (AFEM) with a conventional least squares method has been presented. As a 3D full-wave forward solver, CST Microwave Studio has been used to model and extract both electric field distribution in the region of interest (ROI) and S-parameters of a circular array consisting of 16 monopole antennas. The data has then been fed into a differential inversion scheme to get a qualitative indicator of how the temperature distribution evolves over a course of the cooling process of a heated object. Different regularization techniques within the Tikhonov framework are also discussed, and a balancing principle for optimal choice of the regularization parameter was used to improve the image reconstruction quality of every 2D slice of the final image. Targets are successfully imaged via proposed numerical methods.

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.

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.

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