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Finite Element Exterior Calculus, Homological Techniques, and Applications

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Finite Element Exterior Calculus, Homological Techniques, and Applications Acta Numerica (2006), pp. 1–155 c Cambridge University Press, 2006 doi: 10.1017/S0962492906210018 Printed in the United Kingdom Finite element exterior calculus, homological techniques, and applications Douglas N. Arnold∗ Institute for Mathematics and its Applications and School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected] Richard S. Falk† Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA E-mail: [email protected] Ragnar Winther‡ Centre of Mathematics for Applications and Department of Informatics, University of Oslo, PO Box 1053, 0316 Oslo, Norway E-mail: [email protected] Dedicated to Carme, Rena, and Rita Finite element exterior calculus is an approach to the design and understand- ing 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 discretiza- tions 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 re- vealed 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 Lapla- cian, Maxwell’s equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners. ∗ Supported in part by NSF grant DMS-0411388. † Supported in part by NSF grant DMS03-08347. ‡ Supported by the Norwegian Research Council. 2 D. N. Arnold, R. S. Falk and R. Winther CONTENTS 1 Introduction 2 Part 1: Exterior calculus, finite elements, and homology 2 Exterior algebra and exterior calculus 9 3 Polynomial differential forms and the Koszul complex 28 4 Polynomial differential forms on a simplex 41 5 Finite element differential forms and their cohomology 58 6 Differential forms with values in a vector space 75 Part 2: Applications to discretization of differential equations 7 The Hodge Laplacian 79 8 Eigenvalue problems 94 9 The HΛ projection and Maxwell’s equations 101 10 Preconditioning 107 11 The elasticity equations 121 References 149 1. Introduction The finite element method is one of the greatest advances in numerical computing of the past century. It has become an indispensable tool for sim- ulation of a wide variety of phenomena arising in science and engineering. A tremendous asset of finite elements is that they not only provide a method- ology to develop numerical algorithms for simulation, but also a theoretical framework in which to assess the accuracy of the computed solutions. In this paper we survey and develop the finite element exterior calculus, a new theoretical approach to the design and understanding of finite element dis- cretizations 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 com- patible with the geometric, topological, and algebraic structures which un- derlie well-posedness of the PDE problem being solved. Applications treated here include the finite element discretization of the Hodge Laplacian (which includes many problems as particular cases), Maxwell’s equations of elec- tromagnetism, the equations of elasticity, and elliptic eigenvalue problems, and the construction of preconditioners. To design a finite element method to solve a problem in partial differen- tial equations, the problem is first given a weak or variational formulation which characterizes the solution among all elements of a given space of func- tions on the domain of interest. A finite element method for this problem Finite element exterior calculus 3 proceeds with the construction of a finite-dimensional subspace of the given function space where the solution is sought, and then the specification of a unique element of this subspace as the solution of an appropriate set of equations on this finite-dimensional space. In finite element methods, the subspace is constructed from a triangulation or simplicial decomposition of the given domain, using spaces of polynomials on each simplex, pieced to- gether by a certain assembly process. Because the finite element space so constructed is a subspace of the space where the exact solution is sought, one can consider the difference between the exact and finite element solution and measure it via appropriate norms, seminorms, or functionals. Generally speaking, error bounds can be obtained in terms of three quantities: the ap- proximation error, which measures the error in the best approximation of the exact solution possible within the finite element space, the consistency error, which measures the extent to which the equations used to select the finite element solution from the finite element space reflect the continu- ous problem, and the stability constant, which measures the well-posedness of the finite-dimensional problem. The approximation properties of finite element spaces are well understood, and the consistency of finite element methods is usually easy to control (in fact, for all the methods considered in this paper there is no consistency error in the sense that the exact solution will satisfy the natural extension of the finite element equations to solution space). In marked contrast, the stability of finite element procedures can be very subtle. For many important problems, the development of stable finite element methods remains extremely challenging or even out of reach, and in other cases it is difficult to assess the stability of methods of interest. Lack of stable methods not only puts some important problems beyond the reach of simulation, but has also led to spectacular and costly failures of numerical methods. It should not be surprising that stability is a subtle matter. Establishing stability means proving the well-posedness of the discrete equations, uni- formly in the discretization parameters. Proving the well-posedness of PDE problems is, of course, the central problem of the theory of PDEs. While there are PDE problems for which this is a simple matter, for many im- portant problems it lies deep, and a great deal of mathematics, including analysis, geometry, topology, and algebra, has been developed to estab- lish the well-posedness of various PDE problems. So it is to be expected that a great deal of mathematics bears as well on the stability of numer- ical methods for PDE. An important but insufficiently appreciated point is that approximability and consistency together with well-posedness of the continuous problem do not imply stability. For example, one may consider a PDE problem whose solution is characterized by a variational principle, i.e., as the unique critical point of some functional on some function space, and define a finite element method by seeking a critical point of the same 4 D. N. Arnold, R. S. Falk and R. Winther functional (so there is no consistency error) on a highly accurate finite element space (based on small elements and/or high-order polynomials, so there is arbitrary low approximability error), and yet such a method will very often be unstable and therefore not convergent. Analogously, in the finite difference methodology, one may start with a PDE problem stated in strong form and replace the derivatives in the equation by consistent finite differences, and yet obtain a finite difference method which is unstable. As mentioned, the well-posedness of many PDE problems reflects geo- metrical, algebraic, topological, and homological structures underlying the problem, formalized by exterior calculus, Hodge theory, and homological algebra. In recent years there has been a growing realization that stability of numerical methods can be obtained by designing methods which are com- patible with these structures in the sense that they reproduce or mimic them (not just approximate them) on the discrete level. See, for example, Arnold (2002), and the volume edited by Arnold, Bochev, Lehoucq, Nicolaides and Shashkov (2006a). In the present paper, the compatibility is mostly related to elliptic complexes which are associated with the PDEs under consider- ation, mostly the de Rham complex and its variants and another complex associated with the equations of linear elasticity. Our finite element spaces will arise as the spaces in finite-dimensional subcomplexes which inherit the cohomology and other features of the exact complexes. The inheritance will generally be established by cochain projections: projection operators from the infinite-dimensional spaces in the original elliptic complex which map onto the finite element subspaces and commute with the differential opera- tors of the complex. Thus the main theme of the paper is the development of finite element subcomplexes of certain elliptic differential complexes and cochain projections onto them, and their implications and applications in numerical PDEs. We refer to this theme and the mathematical framework we construct to carry it out, as finite element exterior calculus. We mention some of the computational challenges which motivated the development of finite element exterior calculus and which it has helped to address successfully. These are challenges both of understanding
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