BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 2, April 2010, Pages 281–354 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 discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodge- theoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain pro- jection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham com- plex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.
1. Introduction Numerical algorithms for the solution of partial differential equations are an es- sential tool of the modern world. They are applied in countless ways every day in problems as varied as the design of aircraft, prediction of climate, development of cardiac devices, and modeling of the financial system. Science, engineering, and
Received by the editors June 23, 2009, and, in revised form, August 12, 2009. 2000 Mathematics Subject Classification. Primary: 65N30, 58A14. Key words and phrases. Finite element exterior calculus, exterior calculus, de Rham cohomol- ogy, Hodge theory, Hodge Laplacian, mixed finite elements. The work of the first author was supported in part by NSF grant DMS-0713568. The work of the second author was supported in part by NSF grant DMS-0609755. The work of the third author was supported by the Norwegian Research Council.
c 2010 American Mathematical Society 281
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technology depend not only on efficient and accurate algorithms for approximating the solutions of a vast and diverse array of differential equations which arise in applications, but also on mathematical analysis of the behavior of computed solu- tions, in order to validate the computations, determine the ranges of applicability, compare different algorithms, and point the way to improved numerical methods. Given a partial differential equation (PDE) problem, a numerical algorithm approx- imates the solution by the solution of a finite-dimensional problem which can be implemented and solved on a computer. This discretization depends on a parame- ter (representing, for example, a grid spacing, mesh size, or time step) which can be adjusted to obtain a more accurate approximate solution, at the cost of a larger finite-dimensional problem. The mathematical analysis of the algorithm aims to describe the relationship between the true solution and the numerical solution as the discretization parameter is varied. For example, at its most basic, the theory attempts to establish convergence of the discrete solution to the true solution in an appropriate sense as the discretization parameter tends to zero. Underlying the analysis of numerical methods for PDEs is the realization that convergence depends on consistency and stability of the discretization. Consistency, whose precise definition depends on the particular PDE problem and the type of numerical method, aims to capture the idea that the operators and data defining the discrete problem are appropriately close to those of the true problem for small values of the discretization parameter. The essence of stability is that the discrete problem is well-posed, uniformly with respect to the discretization parameter. Like well- posedness of the PDE problem itself, stability can be very elusive. One might think that well-posedness of the PDE problem, which means invertibility of the operator, together with consistency of the discretization, would imply invertibility of the discrete operator, since invertible operators between a pair of Banach spaces form an open set in the norm topology. But this reasoning is bogus. Consistency is not and cannot be defined to mean norm convergence of the discrete operators to the PDE operator, since the PDE operator, being an invertible operator between infinite- dimensional spaces, is not compact and so is not the norm limit of finite-dimensional operators. In fact, in the first part of the preceding century, a fundamental, and initially unexpected, realization was made: that a consistent discretization of a well-posed problem need not be stable [36, 86, 29]. Only for very special classes of problems and algorithms does well-posedness at the continuous level transfer to stability at the discrete level. In other situations, the development and analysis of stable, consistent algorithms can be a challenging problem, to which a wide array of mathematical techniques has been applied, just as for establishing the well-posedness of PDEs. In this paper we will consider PDEs that are related to differential complexes, for which de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. These are linear elliptic PDEs, but they are a fundamental component of problems arising in many mathematical models, including parabolic, hyperbolic, and nonlinear problems. The finite element exterior calculus, which we develop here, is a theory which was developed to capture the key structures of de Rham cohomology and Hodge theory at the discrete level and to relate the discrete and continuous structures, in order to obtain stable finite element discretizations.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FINITE ELEMENT EXTERIOR CALCULUS 283
1.1. The finite element method. The finite element method, whose develop- ment as an approach to the computer solution of PDEs began over 50 years ago and is still flourishing today, has proven to be one of the most important tech- nologies in numerical PDEs. Finite elements not only provide a methodology to develop numerical algorithms for many different problems, but also a mathematical framework in which to explore their behavior. They are based on a weak or varia- tional form of the PDE problem, and they fall into the class of Galerkin methods, in which the trial space for the weak formulation is replaced by a finite-dimensional subspace to obtain the discrete problem. For a finite element method, this subspace is a space of piecewise polynomials defined by specifying three things: a simplicial decomposition of the domain, and, for each simplex, a space of polynomials called the shape functions, and a set of degrees of freedom for the shape functions, i.e., a basis for their dual space, with each degree of freedom associated to a face of some dimension of the simplex. This allows for efficient implementation of the global piecewise polynomial subspace, with the degrees of freedom determining the degree of interelement continuity. For readers unfamilar with the finite element method, we introduce some basic ideas by considering the approximation of the simplest two-point boundary value problem (1) −u (x)=f(x), −1