Mathematical Foundations of Least-Squares Finite Element Methods

2009 ◽  
pp. 1-33 ◽  
Author(s):  
Pavel B. Bochev ◽  
Max D. Gunzburger
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Finite Element Methods ◽  
Mathematical Foundations

Dispersion Analysis of Stabilized Finite Element Methods for Acoustic Fluid-Structure Interaction

2000 ◽  
Author(s):  
Lonny L. Thompson ◽  
Sridhar Sankar
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Finite Element Methods ◽  
Generalized Least Squares ◽  
Dispersion Analysis ◽  
Stabilized Finite Element Methods ◽  
Stabilized Finite Element ◽  
Stabilized Methods ◽  
Plate Elements ◽  
Acoustic Fluid

Abstract The application of stabilized finite element methods to model the vibration of elastic plates coupled with an acoustic fluid medium is considered. New stabilized methods based on the Hellinger-Reissner variational principle with a generalized least-squares modification are developed which yield improvement in accuracy over the Galerkin and Galerkin Generalized Least Squares (GGLS) finite element methods for both in vacuo and acoustic fluid-loaded Reissner-Mindlin plates. Through judicious selection of design parameters this formulation provides a consistent framework for enhancing the accuracy of mixed Reissner-Mindlin plate elements. Combined with stabilization methods for the acoustic fluid, the method presents a new framework for accurate modeling of acoustic fluid-loaded structures. The technique of complex wave-number dispersion analysis is used to examine the accuracy of the discretized system in the representation of free-waves for fluid-loaded plates. The influence of different finite element approximations for the fluid-loaded plate system are examined and clarified. Improved methods are designed such that the finite element dispersion relations closely match each branch of the complex wavenumber loci for fluid-loaded plates. Comparisons of finite element dispersion relations demonstrate the superiority of the hybrid least-squares (HLS) plate elements combined with stabilized methods for the fluid over standard Galerkin methods with mixed interpolation and shear projection (MITC4) and GGLS methods.

Hybrid First-Order System Least Squares Finite Element Methods with Application to Stokes Equations

2013 ◽  
Vol 51 (4) ◽  
pp. 2214-2237 ◽  
Author(s):  
K. Liu ◽  
T. A. Manteuffel ◽  
S. F. McCormick ◽  
J. W. Ruge ◽  
L. Tang
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Finite Element Methods ◽  
Stokes Equations ◽  
Order System ◽  
First Order
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