scholarly journals NUMERICAL SOLUTION OF THE PROBLEM FOR POISSON’S EQUATION WITH THE USE OF DAUBECHIES WAVELET DISCRETE-CONTINUAL FINITE ELEMENT METHOD

2021 ◽  
Vol 17 (4) ◽  
pp. 123-133
Author(s):  
Marina Mozgaleva ◽  
Pavel Akimov ◽  
Mojtaba Aslami
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Finite Element Approximation ◽  
Poisson’S Equation ◽  
Element Approximation ◽  
Daubechies Wavelet ◽  
Numerical Implementation ◽  
Poisson's Equation ◽  
Element Method

Numerical solution of the problem for Poisson’s equation with the use of Daubechies wavelet discrete continual finite element method (specific version of wavelet-based discrete-continual finite element method) is under consideration in the distinctive paper. The operational initial continual and discrete-continual formulations of the problem are given, several aspects of finite element approximation are considered. Some information about the numerical implementation and an example of analysis are presented.

scholarly journals LOCALIZATION OF SOLUTION OF THE PROBLEM FOR POISSON’S EQUATION WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD

2021 ◽  
Vol 17 (3) ◽  
pp. 157-172
Author(s):  
Marina Mozgaleva ◽  
Pavel Akimov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Basis Functions ◽  
Poisson’S Equation ◽  
Numerical Implementation ◽  
Poisson's Equation ◽  
B Spline ◽  
Element Method

Localization of solution of the problem for Poisson’s equation with the use of B-spline discrete-continual finiteelement method (specificversion of wavelet-based discrete-continual finiteelement method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finiteelement are described, some information about the numerical implementation and an example of analysis are presented.

Analysis of Multi-Crack Growth in Asphalt Pavement Based on Extended Finite Element Method

2012 ◽  
Vol 588-589 ◽  
pp. 1926-1929
Author(s):  
Yu Zhou Sima ◽  
Fu Zhou Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Crack Growth ◽  
Extended Finite Element Method ◽  
Asphalt Pavement ◽  
Finite Element Approximation ◽  
Propagation Path ◽  
Element Approximation ◽  
Extended Finite Element ◽  
Element Method

An extended finite element method (XFEM) for multiple crack growth in asphalt pavement is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite element with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Finally, the propagation path of the cracks in asphalt pavement under different load conditions is presented.

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.

Robust Approximation of Singularly Perturbed Delay Differential Equations by the hp Finite Element Method

2013 ◽  
Vol 13 (1) ◽  
pp. 21-37 ◽  
Author(s):  
Serge Nicaise ◽  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convergence Rates ◽  
Finite Element Approximation ◽  
Interface Layer ◽  
Singularly Perturbed ◽  
Element Approximation ◽  
Smooth Part ◽  
Delay Differential ◽  
Element Method

Abstract. We consider the finite element approximation of the solution to a singularly perturbed second order differential equation with a constant delay. The boundary value problem can be cast as a singularly perturbed transmission problem, whose solution may be decomposed into a smooth part, a boundary layer part, an interior/interface layer part and a remainder. Upon discussing the regularity of each component, we show that under the assumption of analytic input data, the hp version of the finite element method on an appropriately designed mesh yields robust exponential convergence rates. Numerical results illustrating the theory are also included.

Construction of the Beam Element Based Second Generation Wavelet

2011 ◽  
Vol 80-81 ◽  
pp. 532-535
Author(s):  
Yu Min He ◽  
Xi Chen ◽  
Xiao Long Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Second Generation ◽  
Finite Element Approximation ◽  
Beam Element ◽  
Adaptive Finite Element Method ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Second Generation Wavelet ◽  
Element Method

Second generation wavelet (SGW) provides diversity and agility for constructing wavelet besides the multiresolution property. By introducing SGW into finite element method, a series sequence of finite element approximation spaces which are nested and hierarchically expanded can be constructed. The method has high calculation speed and precision and is suited for constructing adaptive algorithm. In this paper, the beam element based SEW is constructed, which set up a basis for the adaptive finite element method based on multiresolution analysis.

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