scholarly journals Superconvergence of a finite element method for linear integro-differential problems

2000 ◽  
Vol 23 (5) ◽  
pp. 343-359 ◽  
Author(s):  
Do Y. Kwak ◽  
Sungyun Lee ◽  
Qian Li
Keyword(s):  
Finite Element Method ◽  
Exact Solution ◽  
Finite Element ◽  
Approximate Solution ◽  
Second Order ◽  
Initial Condition ◽  
First Order ◽  
Superconvergence Results ◽  
Global Superconvergence ◽  
Element Method

We introduce a new way of approximating initial condition to the semidiscrete finite element method for integro-differential equations using any degree of elements. We obtain several superconvergence results for the error between the approximate solution and the Ritz-Volterra projection of the exact solution. Fork>1, we obtain first order gain inLp(2≤p≤∞)norm, second order inW1,p(2≤p≤∞)norm and almost second order inW1,∞norm. Fork=1, we obtain first order gain inW1,p(2≤p≤∞)norms. Further, applying interpolated postprocessing technique to the approximate solution, we get one order global superconvergence between the exact solution and the interpolation of the approximate solution in theLpandW1,p(2≤p≤∞).

The Least-Squares Galerkin Split Finite Element Method for Buoyancy-Driven Flow

2010 ◽  
Author(s):  
Rajeev Kumar ◽  
Brian H. Dennis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Least Squares ◽  
Least Squares Method ◽  
Second Order ◽  
Basis Functions ◽  
Governing Equations ◽  
First Order ◽  
Convection Dominated ◽  
Element Method

The least-squares finite element method (LSFEM), based on minimizing the l2-norm of the residual is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike Galerkin finite element method (GFEM). However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computational requirements that have limited the application of LSFEM to large-scale practical problems. A novel finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal-order basis functions for both pressure and velocity. The method has been successfully applied here to solve complex buoyancy-driven flow with Boussinesq approximation in a square cavity with differentially heated vertical walls using low-order C0 continuous elements.

Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients

2014 ◽  
Vol 12 (11) ◽  
Author(s):  
Xiaofei Guan ◽  
Mingxia Li ◽  
Wenming He ◽  
Zhengwu Jiang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elliptic Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation ◽  
High Degree ◽  
Superconvergence Results ◽  
Element Method

AbstractIn this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x 0 for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h p+1 |ln h|2) and O(h p+2 |ln h|2) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.

THE ROLE OF FINITE ELEMENT METHOD IN CIVIL ENGINEERING

2018 ◽  
Vol 5 (02) ◽  
Author(s):  
Er. Hardik Dhull
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Approximate Solution ◽  
Analytical Method ◽  
Civil Engineering ◽  
The Finite Element Method ◽  
Engineering Problems ◽  
Building Components ◽  
Element Method

The finite element method is a numerical method that is used to find solution of mathematical and engineering problems. It basically deals with partial differential equations. It is very complex for civil engineers to study various structures by using analytical method,so they prefer finite element methods over the analytical methods. As it is an approximate solution, therefore several limitationsare associated in the applicationsin civil engineering due to misinterpretationof analyst. Hence, the main aim of the paper is to study the finite element method in details along with the benefits and limitations of using this method in analysis of building components like beams, frames, trusses, slabs etc.

Sign in / Sign up

Export Citation Format

Share Document