Analysis of least-squares approximations to second-order elliptic problems. I. Finite element method

Suh Yuh Yang

doi:10.1081/NFA-120006702

Analysis of least-squares approximations to second-order elliptic problems. I. Finite element method

Suh Yuh Yang

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4 Scopus citations

Abstract

An L² least-squares finite element method for second-order elliptic problems having non-symmetric diffusion coefficient matrix in two- and three-dimensional bounded domains is proposed and analyzed. The main result is the coercivity estimate of the bilinear form associated with the least-squares functional, which is established by using more direct techniques than that in Ref. [7]. It is proved that the method is not subject to the Ladyzhenskaya-Babuška-Brezzi condition and that the finite element approximation yields a symmetric positive definite linear system with condition number O(h^-2). Optimal error estimates in the H¹(Ω) × H(div; Ω) norm are derived. An equivalent a posteriori error estimator in the above norm is described. Some concluding remarks are also given.

Original language	English
Pages (from-to)	419-432
Number of pages	14
Journal	Numerical Functional Analysis and Optimization
Volume	23
Issue number	3-4
DOIs	https://doi.org/10.1081/NFA-120006702
State	Published - May 2002

Keywords

A posteriori error estimates
A priori error estimates
Elliptic problems
Finite element methods
Least squares

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10.1081/NFA-120006702

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title = "Analysis of least-squares approximations to second-order elliptic problems. I. Finite element method",

abstract = "An L2 least-squares finite element method for second-order elliptic problems having non-symmetric diffusion coefficient matrix in two- and three-dimensional bounded domains is proposed and analyzed. The main result is the coercivity estimate of the bilinear form associated with the least-squares functional, which is established by using more direct techniques than that in Ref. [7]. It is proved that the method is not subject to the Ladyzhenskaya-Babu{\v s}ka-Brezzi condition and that the finite element approximation yields a symmetric positive definite linear system with condition number O(h-2). Optimal error estimates in the H1(Ω) × H(div; Ω) norm are derived. An equivalent a posteriori error estimator in the above norm is described. Some concluding remarks are also given.",

keywords = "A posteriori error estimates, A priori error estimates, Elliptic problems, Finite element methods, Least squares",

author = "Yang, {Suh Yuh}",

note = "Funding Information: This work was supported in part by the National Science Council of Taiwan under grants NSC 89-2115-M-214-002 and NSC 89-2115-M-008-030.",

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TY - JOUR

T1 - Analysis of least-squares approximations to second-order elliptic problems. I. Finite element method

AU - Yang, Suh Yuh

N1 - Funding Information: This work was supported in part by the National Science Council of Taiwan under grants NSC 89-2115-M-214-002 and NSC 89-2115-M-008-030.

PY - 2002/5

Y1 - 2002/5

N2 - An L2 least-squares finite element method for second-order elliptic problems having non-symmetric diffusion coefficient matrix in two- and three-dimensional bounded domains is proposed and analyzed. The main result is the coercivity estimate of the bilinear form associated with the least-squares functional, which is established by using more direct techniques than that in Ref. [7]. It is proved that the method is not subject to the Ladyzhenskaya-Babuška-Brezzi condition and that the finite element approximation yields a symmetric positive definite linear system with condition number O(h-2). Optimal error estimates in the H1(Ω) × H(div; Ω) norm are derived. An equivalent a posteriori error estimator in the above norm is described. Some concluding remarks are also given.

AB - An L2 least-squares finite element method for second-order elliptic problems having non-symmetric diffusion coefficient matrix in two- and three-dimensional bounded domains is proposed and analyzed. The main result is the coercivity estimate of the bilinear form associated with the least-squares functional, which is established by using more direct techniques than that in Ref. [7]. It is proved that the method is not subject to the Ladyzhenskaya-Babuška-Brezzi condition and that the finite element approximation yields a symmetric positive definite linear system with condition number O(h-2). Optimal error estimates in the H1(Ω) × H(div; Ω) norm are derived. An equivalent a posteriori error estimator in the above norm is described. Some concluding remarks are also given.

KW - A posteriori error estimates

KW - A priori error estimates

KW - Elliptic problems

KW - Finite element methods

KW - Least squares

UR - http://www.scopus.com/inward/record.url?scp=0036578936&partnerID=8YFLogxK

U2 - 10.1081/NFA-120006702

DO - 10.1081/NFA-120006702

M3 - 期刊論文

AN - SCOPUS:0036578936

SN - 0163-0563

VL - 23

SP - 419

EP - 432

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

IS - 3-4

ER -

Analysis of least-squares approximations to second-order elliptic problems. I. Finite element method

Abstract

Keywords

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