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This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. BULLETIN (New Series) OF THE
 

Summary: This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.
BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 47, Number 2, April 2010, Pages 281354
S 0273-0979(10)01278-4
Article electronically published on January 25, 2010
FINITE ELEMENT EXTERIOR CALCULUS:
FROM HODGE THEORY TO NUMERICAL STABILITY
DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER
Abstract. This article reports on the confluence of two streams of research,
one emanating from the fields of numerical analysis and scientific computa-
tion, the other from topology and geometry. In it we consider the numerical
discretization of partial differential equations that are related to differential
complexes so that de Rham cohomology and Hodge theory are key tools for
exploring the well-posedness of the continuous problem. The discretization
methods we consider are finite element methods, in which a variational or
weak formulation of the PDE problem is approximated by restricting the trial
subspace to an appropriately constructed piecewise polynomial subspace. Af-
ter a brief introduction to finite element methods, we develop an abstract
Hilbert space framework for analyzing the stability and convergence of such

  

Source: Arnold, Douglas N. - School of Mathematics, University of Minnesota
Falk, Richard S.- Department of Mathematics, Rutgers University
Winther, Ragnar - Department of Informatics & Matematisk institutt, Universitetet i Oslo

 

Collections: Mathematics