Conforming Finite Element Approximations for a Fourth-Order Singular Perturbation Problem

1992 ◽  
Vol 29 (4) ◽  
pp. 1043-1058 ◽  
Author(s):  
Bill Semper
Keyword(s):  
Finite Element ◽  
Singular Perturbation ◽  
Fourth Order ◽  
Singular Perturbation Problem ◽  
Finite Element Approximations ◽  
Perturbation Problem ◽  
Conforming Finite Element

AC0-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem

2018 ◽  
Vol 52 (5) ◽  
pp. 1981-2001 ◽  
Author(s):  
Yuan Bao ◽  
Zhaoliang Meng ◽  
Zhongxuan Luo
Keyword(s):  
Singular Perturbation ◽  
Fourth Order ◽  
Degrees Of Freedom ◽  
Perturbation Parameter ◽  
Singular Perturbation Problem ◽  
Mean Values ◽  
Perturbation Problem ◽  
Convex Quadrilateral ◽  
The Mean ◽  
Quadrilateral Finite Element

In this paper, aC0nonconforming quadrilateral element is proposed to solve the fourth-order elliptic singular perturbation problem. For each convex quadrilateralQ, the shape function space is the union ofS21(Q*) and a bubble space. The degrees of freedom are defined by the values at vertices and midpoints on the edges, and the mean values of integrals of normal derivatives over edges. The local basis functions of our element can be expressed explicitly by a new reference quadrilateral rather than by solving a linear system. It is shown that the method converges uniformly in the perturbation parameter. Lastly, numerical tests verify the convergence analysis.

scholarly journals Convergence Analysis of Incomplete Biquadratic Rectangular Element for Fourth-Order Singular Perturbation Problem on Anisotropic Meshes

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Pingli Xie ◽  
Meng Hu
Keyword(s):  
Singular Perturbation ◽  
Convergence Analysis ◽  
Fourth Order ◽  
Finite Element Approximation ◽  
Perturbation Parameter ◽  
Singular Perturbation Problem ◽  
Element Approximation ◽  
Anisotropic Meshes ◽  
Perturbation Problem ◽  
Rectangular Element

The convergence analysis of a Morley type rectangular element for the fourth-order elliptic singular perturbation problem is considered. A counterexample is provided to show that the element is not uniformly convergent with respect to the perturbation parameter. A modified finite element approximation scheme is used to get convergent results; the corresponding error estimate is presented under anisotropic meshes. Numerical experiments are also carried out to demonstrate the theoretical analysis.

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