Summary: DIFFERENTIAL COMPLEXES AND STABILITY OF FINITE
ELEMENT METHODS I. THE DE RHAM COMPLEX
DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER
Abstract. In this paper we explain the relation between certain piecewise polyno-
mial subcomplexes of the de Rham complex and the stability of mixed finite element
methods for elliptic problems.
Key words. Mixed finite element method, de Rham complex, stability.
AMS(MOS) subject classifications. 65N12, 65N30.
1. Introduction. Many standard finite element methods are based
on extremal variational formulations. Typically, the desired solution is
characterized as the minimum of some functional over an appropriate trial
space of functions, and the discrete solution is then taken to be the mini-
mum of the same functional restricted to a finite dimensional subspace of
the trial space consisting of piecewise polynomials with respect to a trian-
gulation of the domain of interest. For such methods, stability is often a
simple consideration. For mixed finite element methods, which are based on
saddle-point variational principles, the situation is very different: stability
is generally a subtle matter and the development of stable mixed finite ele-
ment methods very challenging. In recent years, a new approach has added
greatly to our understanding of stability of mixed methods and enabled the