Uniform Error Estimates for an Exponentially Fitted Finite Element Method for Singularly Perturbed Elliptic Equations

1999 ◽  
Vol 36 (6) ◽  
pp. 1709-1738 ◽  
Author(s):  
W. Dörfler
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Elliptic Equations ◽  
Singularly Perturbed ◽  
Uniform Error ◽  
Exponentially Fitted ◽  
Uniform Error Estimates ◽  
Element Method

A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number

2017 ◽  
Vol 22 (1) ◽  
pp. 133-156 ◽  
Author(s):  
Yu Du ◽  
Zhimin Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Phase Error ◽  
Three Dimensions ◽  
Wave Number ◽  
Weak Galerkin ◽  
High Wave Number ◽  
Large Wave ◽  
Element Method

AbstractWe study the error analysis of the weak Galerkin finite element method in [24, 38] (WG-FEM) for the Helmholtz problem with large wave number in two and three dimensions. Using a modified duality argument proposed by Zhu and Wu, we obtain the pre-asymptotic error estimates of the WG-FEM. In particular, the error estimates with explicit dependence on the wave numberkare derived. This shows that the pollution error in the brokenH1-norm is bounded byunder mesh conditionk7/2h2≤C0or (kh)2+k(kh)p+1≤C0, which coincides with the phase error of the finite element method obtained by existent dispersion analyses. Herehis the mesh size,pis the order of the approximation space andC0is a constant independent ofkandh. Furthermore, numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the WG-FEM in reducing the pollution effect.

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.

Sign in / Sign up

Export Citation Format

Share Document