Finite element differential forms

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such differential equations by finite element methods, stable discretization depends on the development of spaces of finite element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of order zero, which are simply functions, these two families reduce to one, which is simply the well-known family of Lagrange finite element subspaces of the first order Sobolev space. For forms of degree 1 and of degree n − 1 (where n is the space dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element spaces developed over the last decades. This note reports on some results which form part of the finite element exterior calculus (FEEC) developed in [1], to which we refer for more complete results, details, and further references. FEEC is a new way of looking at finite element spaces used to discretize some of the most fundamental differential operators. It has brought great clarity and unity to the development and analysis of mixed finite elements for a variety of problems, and has enabled major advances in finite elements for elasticity, preconditioning, a posteriori error estimates, and implementation. The fundamental idea of FEEC is to mimic the framework of exterior calculus by developing finite element spaces of differential forms which exactly transfer key geometrical properties of de Rham theory and Hodge theory from the continuous to the discrete level. We recall the basic framework. Let Ω denote a bounded region in R n (for the start it could be a smooth n-dimensional manifold). For each point x of Ω, the tangent space TxΩ is an n-dimensional vector space. If f is a smooth real-valued function of Ω, then df x is a linear map from TxΩ to R. The differential df is a covector field or a 1-form. More generally, an exterior k-form on Ω is a field ω for which ωx is a skew-symmetric k-linear form on TxΩ ×···× TxΩ → R for each x ∈ Ω. We denote the vector space of all smooth k-forms on Ω by Λ k (Ω). The exterior derivative of such a k-form ω is a (k +1 )-form d k ω. It satisfies d k+1 ◦ d k =0 , so the de Rham complex 0 → Λ 0 (Ω) d 0