What are C and h?: Inequalities for the analysis and design of finite element methods

Isaac Harari; Thomas J.R. Hughes

doi:10.1016/0045-7825(92)90162-D

What are C and h? Inequalities for the analysis and design of finite element methods

Isaac Harari^*
, Thomas J.R. Hughes

^*Corresponding author for this work

Stanford University

Research output: Contribution to journal › Article › peer-review

180 Scopus citations

Abstract

Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh-dependent terms in new classes of methods for a variety of applications. Several inquualities of functional analysis are often employed in convergence proofs. In the following, Poincaré-Friedrichs inequalities, inverse estimates and least-squares bounds are characterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of these inequalities are provided, and precise definitions of mesh size that arise naturally in the context of different problems in terms of element geometry are derived.

Original language	English
Pages (from-to)	157-192
Number of pages	36
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	97
Issue number	2
DOIs	https://doi.org/10.1016/0045-7825(92)90162-D
State	Published - Jun 1992
Externally published	Yes

Funding

Funders	Funder number
U.S. Office of Naval Research	N00014-89-K-0027, N00014-88-K-0446

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10.1016/0045-7825(92)90162-D

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title = "What are C and h?: Inequalities for the analysis and design of finite element methods",

abstract = "Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh-dependent terms in new classes of methods for a variety of applications. Several inquualities of functional analysis are often employed in convergence proofs. In the following, Poincar{\'e}-Friedrichs inequalities, inverse estimates and least-squares bounds are characterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of these inequalities are provided, and precise definitions of mesh size that arise naturally in the context of different problems in terms of element geometry are derived.",

author = "Isaac Harari and Hughes, \{Thomas J.R.\}",

note = "Funding Information: This research was supported in part by the U.S. Office of Naval Research under Contracts N00014-89-K-0027 and N00014-88-K-0446. The authors are grateful to Leo Franca, whose ongoing interest in this work provided an impetus for its publication.",

year = "1992",

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AU - Hughes, Thomas J.R.

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PY - 1992/6

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N2 - Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh-dependent terms in new classes of methods for a variety of applications. Several inquualities of functional analysis are often employed in convergence proofs. In the following, Poincaré-Friedrichs inequalities, inverse estimates and least-squares bounds are characterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of these inequalities are provided, and precise definitions of mesh size that arise naturally in the context of different problems in terms of element geometry are derived.

AB - Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh-dependent terms in new classes of methods for a variety of applications. Several inquualities of functional analysis are often employed in convergence proofs. In the following, Poincaré-Friedrichs inequalities, inverse estimates and least-squares bounds are characterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of these inequalities are provided, and precise definitions of mesh size that arise naturally in the context of different problems in terms of element geometry are derived.

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ER -

What are C and h? Inequalities for the analysis and design of finite element methods

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