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Mathematics of Computation

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Weak coupling of solutions of first-order least-squares method


Author: Jaeun Ku
Journal: Math. Comp. 77 (2008), 1323-1332
MSC (2000): Primary 65N30, 65N15
DOI: https://doi.org/10.1090/S0025-5718-08-02062-0
Published electronically: January 22, 2008
MathSciNet review: 2398770
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Abstract: A theoretical analysis of a first-order least-squares finite element method for second-order self-adjoint elliptic problems is presented. We investigate the coupling effect of the approximate solutions $ u_h$ for the primary function $ u$ and $ \boldsymbol{\sigma}_h$ for the flux $ \boldsymbol{\sigma}=-\mathcal A\nabla u$. We prove that the accuracy of the approximate solution $ u_h$ for the primary function $ u$ is weakly affected by the flux $ \boldsymbol{\sigma}=-\mathcal A\nabla u$. That is, the bound for $ \Vert u-u_h\Vert _1$ is dependent on $ \boldsymbol{\sigma}$, but only through the best approximation for $ \boldsymbol{\sigma}$ multiplied by a factor of meshsize $ h$. Similarly, we provide that the bound for $ \Vert\boldsymbol{\sigma}-\boldsymbol{\sigma}_h\Vert _{H(div)}$ is dependent on $ u$, but only through the best approximation for $ u$ multiplied by a factor of the meshsize $ h$. This weak coupling is not true for the non-selfadjoint case. We provide the numerical experiment supporting the theorems in this paper.


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Additional Information

Jaeun Ku
Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067
Address at time of publication: Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, Oklahoma 74078-1058
Email: jku@math.okstate.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02062-0
Keywords: Least-squares, finite element methods, coupling
Received by editor(s): November 20, 2006
Received by editor(s) in revised form: April 16, 2007
Published electronically: January 22, 2008
Additional Notes: This research was supported in part by NSF grant DMS-0071412.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.