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A nonconforming mixed finite element for second‐order elliptic problems
Author(s) -
Farhloul Mohamed,
Fortin And Michel
Publication year - 1997
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199709)13:5<445::aid-num1>3.0.co;2-j
Subject(s) - mathematics , finite element method , piecewise , equivalence (formal languages) , mixed finite element method , quadratic equation , order (exchange) , partial derivative , convergence (economics) , partial differential equation , extended finite element method , mathematical analysis , pure mathematics , geometry , physics , finance , economics , thermodynamics , economic growth
In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second‐order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so‐called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505–520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445–457, 1997