Uniform error estimates of finite element method for a 3D semilinear elliptic problem with small viscosity

2018 ◽  
Vol 463 (2) ◽  
pp. 869-879
Author(s):  
Yarong Zhang ◽  
Yinnian He ◽  
Hongbin Chen
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Elliptic Problem ◽  
Uniform Error ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Small Viscosity ◽  
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.

scholarly journals A mixed finite element method for a sixth-order elliptic problem

2017 ◽  
Vol 39 (1) ◽  
pp. 374-397 ◽  
Author(s):  
Jérôme Droniou ◽  
Muhammad Ilyas ◽  
Bishnu P Lamichhane ◽  
Glen E Wheeler
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elliptic Problem ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Sixth Order ◽  
Order Elliptic Problem ◽  
Element Method
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