An asymptotic finite element method for singularly perturbed third and fourth order ordinary differential equations with discontinuous source term

2007 ◽  
Vol 191 (2) ◽  
pp. 372-380 ◽  
Author(s):  
A. Ramesh Babu ◽  
N. Ramanujam
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Source Term ◽  
Singularly Perturbed ◽  
Discontinuous Source Term ◽  
Element Method

A Parameter Robust Finite Element Method for Fourth Order Singularly Perturbed Problems

2017 ◽  
Vol 17 (2) ◽  
pp. 337-349 ◽  
Author(s):  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourth Order ◽  
Hermite Polynomials ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
The Finite Element Method ◽  
Singularly Perturbed Problems ◽  
Exponentially Graded ◽  
Element Method

AbstractWe consider fourth order singularly perturbed problems in one-dimension and the approximation of their solution by the h version of the finite element method. In particular, we use piecewise Hermite polynomials of degree ${p\geq 3}$ defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error is measured in both the energy norm and a stronger, ‘balanced’ norm. Finally, we illustrate our theoretical findings through numerical computations, including a comparison with another scheme from the literature.

Correct Discrete-Continual Finite Element Method of Structural Analysis Based on Precise Analytical Solutions of Resulting Multipoint Boundary Problems for Systems of Ordinary Differential Equations

2012 ◽  
Vol 204-208 ◽  
pp. 4502-4505 ◽  
Author(s):  
Pavel A. Akimov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Structural Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Solutions ◽  
Computational Stability ◽  
Boundary Problems ◽  
Multipoint Boundary ◽  
Element Method

The distinctive paper is devoted to correct discrete-continual finite element method (DCFEM) of structural analysis based on precise analytical solutions of resulting multipoint boundary problems for systems of ordinary differential equations with piecewise-constant coefficients. Corresponding semianalytical (discrete-continual) formulations are contemporary mathematical models which currently becoming available for computer realization. Major peculiarities of DCFEM include uni-versality, computer-oriented algorithm involving theory of distributions, computational stability, optimal conditionality of resulting systems and partial Jordan decompositions of matrices of coeffi-cients, eliminating necessity of calculation of root vectors.

Sign in / Sign up

Export Citation Format

Share Document