Richard S. Falk

Douglas N. Arnold, Richard S. Falk, Ragnar Winther

We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart-Thomas, Brezzi-Douglas-Marini, and Nedelec spaces. In the present paper, we derive geometric decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis.

Douglas N. Arnold, Richard S. Falk, Jay Gopalakrishnan

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

Richard S. Falk, Ragnar Winther

We construct projections from the space of differential k-forms which belong to L2 and whose exterior derivative also belongs to L2, to finite dimensional subspaces of piecewise polynomial differential forms defined on a simplicial mesh. These projections have the properties that they commute with the exterior derivative and are bounded independent of the mesh size. Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Clement interpolant.

Douglas N. Arnold, Richard S. Falk, Johnny Guzmán et al.

In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferential of the form as the triangulation is refined. In 1991, Smits proved this to be the case for the combinatorial codifferential applied to 1-forms in two dimensions under the additional assumption that the initial triangulation is refined in a completely regular fashion, by dividing each triangle into four similar triangles. In this paper we extend Smits's result to arbitrary dimensions, showing that the combinatorial codifferential on 1-forms is consistent if the triangulations are uniform or piecewise uniform in a certain precise sense. We also show that this restriction on the triangulations is needed, giving a counterexample in which a different regular refinement procedure, namely Whitney's standard subdivision, is used. Further, we show by numerical example that for 2-forms in three dimensions, the combinatorial codifferential is not consistent even for the most regular subdivision process.

Douglas N. Arnold, Richard S. Falk, Ragnar Winther

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, 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 the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, 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 complex 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 to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.

Douglas N. Arnold, Richard S. Falk, Ragnar Winther

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger--Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.