Finite Element Exterior Calculus Research Articles

Serendipity and Tensor Product Affine Pyramid Finite Elements

Using the language of finite element exterior calculus, we define two families of H 1 -conforming finite element spaces over pyramids with a parallelogram base. The first family has matching polynomial traces with tensor product elements on the base while the second has matching polynomial traces with serendipity elements on the base. The second family is new to the literature and provides a robust approach for linking between Lagrange elements on tetrahedra and serendipity elements on affinely-mapped cubes while preserving continuity and approximation properties. We define shape functions and degrees of freedom for each family and prove unisolvence and polynomial reproduction results.

Finite element exterior calculus with lower-order terms

The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they frequently show up perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is well understood, but that in the vector case is not. In this paper, we first show that, surprisingly, for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further, our results imply many previous results for the scalar problem and thus unify them all under the FEEC framework.

Compatible-strain mixed finite element methods for 2D compressible nonlinear elasticity

In this paper, using the Hilbert complexes of nonlinear elasticity, the approximation theory for Hilbert complexes, and the finite element exterior calculus, we introduce a new class of mixed finite element methods for 2D nonlinear elasticity–compatible-strain mixed finite element methods (CSFEM). We consider a Hu–Washizu-type mixed formulation and choose the displacement, the displacement gradient, and the first Piola–Kirchhoff stress tensor as independent unknowns. We use the underlying spaces of the Hilbert complexes as the solution and test spaces. We discretize the Hilbert complexes and introduce a new class of mixed finite element methods for nonlinear elasticity by using the underlying finite element spaces of the discrete Hilbert complexes. This automatically enforces the trial spaces of the displacement gradient to satisfy the classical Hadamard jump condition, which is a necessary condition for the compatibility of non-smooth displacement gradients. The underlying finite element spaces of CSFEMs are the tensorial analogues of the standard Nédélec and Raviart–Thomas elements of vector fields. These spaces respect the global topologies of the domains in the sense that they can reproduce certain topological properties of the bodies regardless of the refinement level of meshes. By solving several numerical examples, we demonstrate that CSFEMs have a good performance for bending problems, in the near incompressible regime, and for bodies with complex geometries. CSFEMs are capable of accurately approximating stresses and perform well in problems that standard enhanced strain methods suffer from the hourglass instability. Moreover, CSFEMs provide a convenient framework for modeling inhomogeneities.

Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes

We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart-Thomas, and Brezzi-Douglas-Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

Complexes of Discrete Distributional Differential Forms and Their Homology Theory

Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus, we generalize a notion of Braess and Schoberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. Poincare–Friedrichs-type inequalities will be studied in a subsequent contribution.

The Abstract Hodge--Dirac Operator and Its Stable Discretization

This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge--Dirac operator, which is a square root of the abstract Hodge--Laplace operator considered by Arnold, Falk, and Winther [Bull. Amer. Math. Soc., 47 (2010), pp. 281--354]. Dirac-type operators are central to the field of Clifford analysis, where recently there has been considerable interest in their discretization. We prove a priori stability and convergence estimates, and show that several of the results in finite element exterior calculus can be recovered as corollaries of these new estimates. (A corrected version is attached.)

On high order finite element spaces of differential forms

We show how the high order finite element spaces of differential forms due to Raviart-Thomas-Nedelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of Arnold-Falk-Winther. Based on observations by Bossavit, we provide new low order degrees of freedom. As an alternative to existing choices of bases, we provide canonical resolutions in terms of scalar polynomials and Whitney forms.

Double complexes and local cochain projections.

The construction of projection operators, which commute with the exterior derivative and at the same time are bounded in the proper Sobolev spaces, represents a key tool in the recent stability analysis of finite element exterior calculus. These so-called bounded cochain projections have been constructed by combining a smoothing operator and the unbounded canonical projections defined by the degrees of freedom. However, an undesired property of these bounded projections is that, in contrast to the canonical projections, they are nonlocal. The purpose of this article is to discuss a recent alternative construction of bounded cochain projections, which also are local. A key tool for the new construction is the structure of a double complex, resembling the Čech-de Rham double complex of algebraic topology. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 541–551, 2015

A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.

Low-Complexity Finite Element Algorithms for the de Rham Complex on Simplices

We combine recently developed finite element algorithms based on Bernstein polynomials [M. Ainsworth, G. Andriamaro, and O. Davydov, SIAM J. Sci. Comput., 33 (2011), pp. 3087--3109; R. C. Kirby, Numer. Math., 117 (2011), pp. 631--652] with the explicit basis construction of the finite element exterior calculus [D. N. Arnold, R. S. Falk, and R. Winther, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1660--1672] to give a family of algorithms for the de Rham complex on simplices that achieves stiffness matrix construction and matrix-free action in optimal complexity. These algorithms are based on realizing the exterior calculus bases as short combinations of Bernstein polynomials. Numerical results confirm optimal scaling of the algorithms and favorable comparison with FEniCS at high polynomial order as well. Additional empirical studies show that very high accuracy is achieved in the mixed discretization of the Poisson equation and the Maxwell eigenvalue problem as the polynomial degree increases.

Finite element differential forms on cubical meshes

We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new H ( c u r l ) H(\mathrm {curl}) and H ( d i v ) H(\mathrm {div}) finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus.

A finite element exterior calculus framework for the rotating shallow-water equations

We describe discretisations of the shallow-water equations on the sphere using the framework of finite element exterior calculus, which are extensions of the mimetic finite difference framework presented in Ringler (2010) [11]. The exterior calculus notation provides a guide to which finite element spaces should be used for which physical variables, and unifies a number of desirable properties. We present two formulations: a “primal” formulation in which the finite element spaces are defined on a single mesh, and a “primal–dual” formulation in which finite element spaces on a dual mesh are also used. Both formulations have velocity and layer depth as prognostic variables, but the exterior calculus framework leads to a conserved diagnostic potential vorticity. In both formulations we show how to construct discretisations that have mass-consistent (constant potential vorticity stays constant), stable and oscillation-free potential vorticity advection.

PyDEC

This article describes the algorithms, features, and implementation of PyDEC, a Python library for computations related to the discretization of exterior calculus. PyDEC facilitates inquiry into both physical problems on manifolds as well as purely topological problems on abstract complexes. We describe efficient algorithms for constructing the operators and objects that arise in discrete exterior calculus, lowest-order finite element exterior calculus, and in related topological problems. Our algorithms are formulated in terms of high-level matrix operations which extend to arbitrary dimension. As a result, our implementations map well to the facilities of numerical libraries such as NumPy and SciPy. The availability of such libraries makes Python suitable for prototyping numerical methods. We demonstrate how PyDEC is used to solve physical and topological problems through several concise examples.

Mixed finite elements for numerical weather prediction

We show how mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite difference methods. There are a few major advantages: the mixed finite element methods do not require an orthogonal grid, and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation. These methods preserve several properties of the C-grid method when applied to linear barotropic wave propagation, namely: (a) energy conservation, (b) mass conservation, (c) no spurious pressure modes, and (d) steady geostrophic modes on the f-plane. We explain how these properties are preserved, and describe two examples that can be used on pseudo-uniform grids: the recently-developed modified RTk-Q(k-1) element pairs on quadrilaterals and the BDFM1-P1DG element pair on triangles. All of these mixed finite element methods have an exact 2:1 ratio of velocity degrees of freedom to pressure degrees of freedom. Finally we illustrate the properties with some numerical examples.

Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces

A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational problem on a finite-dimensional subcomplex. In a seemingly unrelated research direction, Dziuk [Lect Notes in Math, vol 1357 (1988)] analyzed a class of nodal finite elements for the Laplace-Beltrami equation on smooth 2-surfaces approximated by a piecewise-linear triangulation; Demlow later extended this analysis [SIAM J Numer Anal, 47 (2009)] to 3-surfaces, as well as to higher-order surface approximation. In this article, we bring these lines of research together, first developing a framework for the analysis of variational crimes in abstract Hilbert complexes, and then applying this abstract framework to the setting of finite element exterior calculus on hypersurfaces. Our framework extends the work of Arnold, Falk, and Winther to problems that violate their subcomplex assumption, allowing for the extension of finite element exterior calculus to approximate domains, most notably the Hodge-de Rham complex on approximate manifolds. As an application of the latter, we recover Dziuk's and Demlow's a priori estimates for 2- and 3-surfaces, demonstrating that surface finite element methods can be analyzed completely within this abstract framework. Moreover, our results generalize these earlier estimates dramatically, extending them from nodal finite elements for Laplace-Beltrami to mixed finite elements for the Hodge Laplacian, and from 2- and 3-dimensional hypersurfaces to those of arbitrary dimension. By developing this analytical framework using a combination of general tools from differential geometry and functional analysis, we are led to a more geometric analysis of surface finite element methods, whereby the main results become more transparent.

Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation

Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010), 281-354] that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract language of Hilbert complexes. In another recent article [arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzing variational crimes (a la Strang) on Hilbert complexes. In particular, this gave a treatment of finite element exterior calculus on manifolds, generalizing techniques from surface finite element methods and recovering earlier a priori estimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk [Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J. Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, we extend the Hilbert complex framework in a second distinct direction: to the study of semilinear mixed problems. We do this, first, by introducing an operator-theoretic reformulation of the linear mixed problem, so that the semilinear problem can be expressed as an abstract Hammerstein equation. This allows us to obtain, for semilinear problems, a priori solution estimates and error estimates that reduce to the Arnold-Falk-Winther results in the linear case. We also consider the impact of variational crimes, extending the results of our previous article to these semilinear problems. As an immediate application, this new framework allows for mixed finite element methods to be applied to semilinear problems on surfaces.

Finite element exterior calculus: from Hodge theory to numerical stability

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.

Smoothed projections in finite element exterior calculus

The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, has proved to be an effective tool in finite element exterior calculus. The advantage of these operators is that they are $L^2$ bounded projections, and still they commute with the exterior derivative. In the present paper we generalize the construction of these smoothed projections, such that also non-quasi-uniform meshes and essential boundary conditions are covered. The new tool introduced here is a space-dependent smoothing operator that commutes with the exterior derivative.

Finite element differential forms

AbstractA 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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite element exterior calculus, homological techniques, and applications

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.