scholarly journals Uniform Shift Estimates for Transmission Problems and Optimal Rates of Convergence for the Parametric Finite Element Method

2013 ◽  
pp. 12-23 ◽  
Author(s):  
Hengguang Li ◽  
Victor Nistor ◽  
Yu Qiao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Rates Of Convergence ◽  
Transmission Problems ◽  
Optimal Rates Of Convergence ◽  
Element Method

scholarly journals THE HP-VERSION OF THE STREAMLINE DIFFUSION FINITE ELEMENT METHOD IN TWO SPACE DIMENSIONS

2001 ◽  
Vol 11 (02) ◽  
pp. 301-337 ◽  
Author(s):  
K. GERDES ◽  
J. M. MELENK ◽  
C. SCHWAB ◽  
D. SCHÖTZAU
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Rates Of Convergence ◽  
Diffusion Problem ◽  
Convection Diffusion ◽  
Consistency Results ◽  
Two Space Dimensions ◽  
Streamline Diffusion ◽  
Exponential Rates ◽  
Element Method

The Streamline Diffusion Finite Element Method (SDFEM) for a two-dimensional convection–diffusion problem is analyzed in the context of the hp-version of the Finite Element Method (FEM). It is proved that the appropriate choice of the SDFEM parameters leads to stable methods on the class of "boundary layer meshes", which may contain anisotropic needle elements of arbitrarily high aspect ratio. Consistency results show that the use of such meshes can resolve layer components present in the solutions at robust exponential rates of convergence. We confirm these theoretical results in a series of numerical examples.

scholarly journals A Consistent Immersed Finite Element Method for the Interface Elasticity Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Sangwon Jin ◽  
Do Y. Kwak ◽  
Daehyeon Kyeong
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Rates Of Convergence ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Immersed Finite Element Method ◽  
Elasticity Problems ◽  
Interface Elasticity ◽  
Nonconforming Finite Element Methods ◽  
Element Method

We propose a new scheme for elasticity problems having discontinuity in the coefficients. In the previous work (Kwak et al., 2014), the authors suggested a method for solving such problems by finite element method using nonfitted grids. The proposed method is based on theP1-nonconforming finite element methods with stabilizing terms. In this work, we modify the method by adding the consistency terms, so that the estimates of consistency terms are not necessary. We show optimal error estimates inH1and divergence norms under minimal assumptions. Various numerical experiments also show optimal rates of convergence.

scholarly journals A generalized finite element method for problems with sign-changing coefficients

2021 ◽  
Author(s):  
Théophile Chaumont-Frelet ◽  
Barbara Verfürth
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convergence Rates ◽  
Linear Convergence ◽  
Generalized Finite Element Method ◽  
Transmission Problems ◽  
Optimal Linear ◽  
Generalized Finite Element ◽  
Positive Materials ◽  
Element Method

Problems with sign-changing coefficients occur, for instance, in the study of transmission problems with metamaterials. In this work, we present and analyze a generalized finite element method in the spirit of the Localized Orthogonal Decomposition, that is especially efficient when the negative and positive materials exhibit multiscale features. We derive optimal linear convergence in the energy norm independently of the potentially low regularity of the exact solution. Numerical experiments illustrate the theoretical convergence rates and show the applicability of the method for a large class of sign-changing diffusion problems.

A stabilizer free spatial weak Galerkin finite element methods for time-dependent convection-diffusion equations

2021 ◽  
pp. 1-16
Author(s):  
Ahmed Al-Taweel ◽  
Saqib Hussain ◽  
Xiaoshen Wang ◽  
Mohammed Cheichan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Rates Of Convergence ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Weak Galerkin ◽  
Simple Formulation ◽  
Spatial Direction ◽  
Convection Diffusion ◽  
Element Method

In this paper, we propose a stabilizer free spatial weak Galerkin (SFSWG) finite element method for solving time-dependent convection diffusion equations based on weak form Eq. (4). SFSWG method in spatial direction and Euler difference operator Eq. (37) in temporal direction are used. The main reason for using the SFSWG method is because of its simple formulation that makes this algorithm more efficient and its implementation easier. The optimal rates of convergence of 𝒪⁢(hk) and 𝒪⁢(hk+1) in a discrete H1 and L2 norms, respectively, are obtained under certain conditions if polynomial spaces (Pk⁢(K),Pk⁢(e),[Pj⁢(K)]2) are used in the SFSWG finite element method. Numerical experiments are performed to verify the effectiveness and accuracy of the SFSWG method.

HAMILTONIAN AS ERROR INDICATOR IN THE P-VERSION OF FINITE ELEMENT METHOD

2010 ◽  
Vol 34 (2) ◽  
pp. 215-223
Author(s):  
Yong-Lin Kuo ◽  
William L. Cleghorn ◽  
Kamran Behdinan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Rates Of Convergence ◽  
Computational Time ◽  
Governing Equations ◽  
Accuracy Enhancement ◽  
Error Indicator ◽  
Energy Error ◽  
Interpolation Polynomials ◽  
Element Method

This paper presents the Hamiltonian-based error analysis applied to two-dimensional elasostatic problems. The accuracy enhancement is achieved by using the p-version of finite element method. The results show that the Hamiltonian error has faster rates of convergence at lower order of interpolation polynomials to compare with the energy error, and the Hamiltonian error clearly indicates great error reductions at a certain polynomial order. This can not only obtain an accurate enough solution but also save extra computational time. Another strategy is presented by computing the residual of the Hamiltonian-based governing equations. Relative values of residuals between elements can provide an index of selecting the best polynomial orders. Illustrative examples show the validities of the two approaches.

Non-conforming Computational Methods for Mixed Elasticity Problems

2003 ◽  
Vol 3 (1) ◽  
pp. 23-34
Author(s):  
Faker Ben Belgacem ◽  
Lawrence K. Chilton ◽  
Padmanabhan Seshaiyer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Elements ◽  
Rates Of Convergence ◽  
Modeling Methodology ◽  
Conforming Finite Element ◽  
Elasticity Problems ◽  
Conforming Finite Element Method ◽  
Incompressible Elasticity ◽  
Element Method

AbstractIn this paper, we present a non-conforming hp computational modeling methodology for solving elasticity problems. We consider the incompressible elasticity model formulated as a mixed displacement-pressure problem on a global domain which is partitioned into several local subdomains. The approximation within each local subdomain is designed using div-stable hp-mixed finite elements. The displacement is computed in a mortared space while the pressure is not subjected to any constraints across the interfaces. Our computational results demonstrate that the non-conforming finite element method presented for the elasticity problem satisfies similar rates of convergence as the conforming finite element method, in the presence of various h-version and p-version discretizations.

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