Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are revealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Laplacian, Maxwell’s equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

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 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. After 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 projection 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 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 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.

We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces H(div) and H(curl ) in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products. Subject Classification (1991): 65N55, 65N30 Mathematics

Abstract. 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.

Abstract. The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n − 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).Mathematics Subject Classification. 65N30, 65N12, 65N15, 76R50.

Abstract. We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r + 1 in L p and order r in W 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

This paper is dedicated to Jim Douglas, Jr., on the occasion of his 60th birthday.

Abstract. We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to a space of vector fields on each convex quadrilateral element via the Piola transform associated to a bilinear isomorphism of the square onto the element. For affine isomorphisms, a necessary and sufficient condition for approximation of order r + 1 in L 2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, the situation is more complicated and we give a precise characterization of what is needed for optimal order L 2 -approximation of the function and of its divergence. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of H(div). We also derive new estimates for approximation by quadrilateral Raviart-Thomas elements (requiring less regularity) and propose a new quadrilateral finite element space which provides optimal order approximation in H(div). Finally, we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of Poisson's equation.