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Volume 1, Issue 1
Least-Squares Finite Element Methods for First-Order Elliptic Systems

Pavel Bochev

Int. J. Numer. Anal. Mod., 1 (2004), pp. 49-64.

Published online: 2004-01

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  • Abstract

Least-squares principles use artificial "energy" functionals to provide a Rayleigh-Ritz-like setting for the finite element method. These functionals are defined in terms of PDE’s residuals and are not unique. We show that viable methods result from reconciliation of a mathematical setting dictated by the norm-equivalence of least-squares functionals with practicality constraints dictated by the algorithmic design. We identify four universal patterns that arise in this process and develop this paradigm for first-order ADN elliptic systems. Special attention is paid to the effects that each discretization pattern has on the computational and analytic properties of finite element methods, including error estimates, conditioning of the algebraic systems and the existence of efficient preconditioners.

  • Keywords

finite elements, least-squares, first-order elliptic systems.

  • AMS Subject Headings

65F10, 65F30

  • Copyright

COPYRIGHT: © Global Science Press

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  • BibTex
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  • TXT
@Article{IJNAM-1-49, author = {Bochev , Pavel}, title = {Least-Squares Finite Element Methods for First-Order Elliptic Systems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2004}, volume = {1}, number = {1}, pages = {49--64}, abstract = {

Least-squares principles use artificial "energy" functionals to provide a Rayleigh-Ritz-like setting for the finite element method. These functionals are defined in terms of PDE’s residuals and are not unique. We show that viable methods result from reconciliation of a mathematical setting dictated by the norm-equivalence of least-squares functionals with practicality constraints dictated by the algorithmic design. We identify four universal patterns that arise in this process and develop this paradigm for first-order ADN elliptic systems. Special attention is paid to the effects that each discretization pattern has on the computational and analytic properties of finite element methods, including error estimates, conditioning of the algebraic systems and the existence of efficient preconditioners.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/965.html} }
TY - JOUR T1 - Least-Squares Finite Element Methods for First-Order Elliptic Systems AU - Bochev , Pavel JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 49 EP - 64 PY - 2004 DA - 2004/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/965.html KW - finite elements, least-squares, first-order elliptic systems. AB -

Least-squares principles use artificial "energy" functionals to provide a Rayleigh-Ritz-like setting for the finite element method. These functionals are defined in terms of PDE’s residuals and are not unique. We show that viable methods result from reconciliation of a mathematical setting dictated by the norm-equivalence of least-squares functionals with practicality constraints dictated by the algorithmic design. We identify four universal patterns that arise in this process and develop this paradigm for first-order ADN elliptic systems. Special attention is paid to the effects that each discretization pattern has on the computational and analytic properties of finite element methods, including error estimates, conditioning of the algebraic systems and the existence of efficient preconditioners.

Bochev , Pavel. (2004). Least-Squares Finite Element Methods for First-Order Elliptic Systems. International Journal of Numerical Analysis and Modeling. 1 (1). 49-64. doi:
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