Paper Info
Title
Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry.
Abstract
This paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the L-2 norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the norm of H(div; Omega) for the stress, of H-1(Omega) for the displacement/velocity, and of L-2(Omega) for the rotation/vorticity and the pressure. This immediately implies optimal error estimates in the energy norm for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart-Thomas finite element spaces are used to approximate the stress tensor. Through a refined duality argument, an optimal L-2 norm error estimates for the displacement/velocity are also established. Finally, numerical results for a Cook's membrane problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit.
Year
DOI
Venue
2019
10.1515/cmam-2018-0255
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
Keywords
Field
DocType
Least-squares,Linear Elasticity,Stokes Equations
Least squares,Mathematical analysis,Elasticity (economics),Mathematics
Journal
Volume
Issue
ISSN
19
SP3
1609-4840
Citations 
PageRank 
References 
0
0.34
0
Authors
3
Name
Order
Citations
PageRank
Fleurianne Bertrand110.70
zhiqiang cai234478.81
Eun Young Park300.34