An abstract Hodge Dirac operator and its stable discretization

Transcription

1 An abstract Hodge Dirac operator and its stable discretization Paul Leopardi Mathematical Sciences Institute, Australian National University. For presentation at ICCA, Tartu,. Joint work with Ari Stern, Washington University in St. Louis. August

2 Introduction Acknowledgements Paula Cerejeiras and Uwe Kähler (Aveiro, Snorre Christiansen and Ragnar Winther (Oslo, Michael Holst (UCSD, Bishnu Lamichhane (Newcastle, Lashi Bandara and Alan McIntosh (ANU. Australian Research Council. Australian Mathematical Sciences Institute. Australian National University.

3 Introduction Subjects with parallel + year histories Finite Element Method M. Zlámal, On the finite element method. Numer. Math.,, 9, pp Clifford analysis D. Hestenes, Multivector calculus. J. Math. Anal. and Appl., :, 9, pp. -.

4 Introduction Topics Finite Element Exterior Calculus Discretization of the Hodge Dirac operator Discretization of the Hodge Laplacian Numerical examples Further considerations

5 Finite Element Exterior Calculus Finite Element Method The Finite Element Method solves boundary value problems based on partial differential equations. The original problem in a Hilbert space of functions is put into variational form, and is mapped into a problem defined on a finite dimensional function space, whose basis consists of functions supported in small regions, such as simplices. (Iserles 99; Braess.

6 Finite Element Exterior Calculus Finite element exterior calculus (FEEC FEEC is based on the Finite Element Method over Hilbert complexes. These are cochain complexes where the relevant vector spaces are Hilbert spaces. For the de Rham complex, FEEC uses Hodge decomposition, the exterior derivative and differential forms. The numerical stability of the FEEC discretization depends on the existence of a bounded cochain projection from a Hilbert complex to a subcomplex. FEEC uses smoothed projections to obtain this numerical stability. (Arnold, Falk and Winther, ; Christiansen and Winther

7 Discretization of the Hodge Dirac operator An abstract Hodge Dirac problem: setting Let d be a closed, densely defined nilpotent operator on the Hilbert space W, with domain V, and closed range B. In V, we use the inner product u, v V := u, v + du, dv. We have the orthogonal Hodge decomposition W = B H B, u = u B u H u B, u W. where Z = B H is the null space of d.

8 Discretization of the Hodge Dirac operator Example: Hilbert complexes If (M, g is an oriented, compact Riemannian manifold, then each space of smooth k-forms has an L -inner product, u, v L Ω k (M = u, v g vol g = u g v. M This gives an adjoint operator d k : Ωk (M Ω k (M for each k, e.g. M The Hodge decomposition says that each f L Ω k (M can be orthogonally decomposed as f = dα + d β + γ, where dγ =, d γ =.

9 Discretization of the Hodge Dirac operator An abstract Hodge Dirac problem We want to solve the problem Du = f f H, where D := d + d is the abstract Hodge Dirac operator. Consider the following mixed variational problem: Find (u, p V H satisfying du, v + u, dv + p, v = f, v, v V, ( u, q =, q H. To show that this problem is well-posed, it suffices to prove the inf-sup condition for the symmetric bilinear form on V H. B(u, p; v, q := du, v + u, dv + p, v + u, q

10 Discretization of the Hodge Dirac operator The problem is well-posed Theorem. (LS There exists a constant γ >, depending only on the Poincaré constant c P, such that for all non-zero (u, p V H, there exists a non-zero (v, q V H satisfying B(u, p; v, q γ( u V + p ( v V + q. Proof. (hint. Consider the test functions v := ρ + p + du, q := u H, where ρ Z is the unique element such that dρ = u B.

11 Discretization of the Hodge Dirac operator A corresponding discrete problem Suppose V h V is a Hilbert subspace, with a bounded projection π h : V V h, such that π h d = dπ h. Consider the discrete problem: Find (u h, p h V h H h satisfying du h, v h + u h, dv h + p h, v h = f, v h, v h V h, ( u h, q h =, q h H h. This problem is well-posed, with a discrete inf-sup condition, where the constant γ h depends only on c P and the norm π h.

12 Discretization of the Hodge Dirac operator An error estimate Theorem. (LS Let (u, p be the solution to ( and (u h, p h be the solution to (. If the projections π h are V -bounded uniformly, independently of h, then the error can be estimated by u u h V + p p h ( C inf u v v V V + inf p q h q V V + µ inf P B u v h v V V h where µ := ( π h P H.,

13 Discretization of the Hodge Laplacian The Hodge Laplace problem The abstract Hodge Laplace operator is L = D = dd + d d, defined on the domain D(L = D (V V V V with kernel N (L = N (D = H. The Hodge Laplace problem is: Given f W, find (u, p ( D(L N (L N (L such that Lu + p = f. To solve this, we can solve Dw + p = f, and then solve Du = w.

14 Discretization of the Hodge Laplacian The mixed variational form The mixed variational form of the Hodge Laplace problem is: Find (σ, u, p V V H such that σ, τ u, dτ =, τ V, dσ, v + du, dv + p, v = f, v, v V, u, q =, q H. (

15 Discretization of the Hodge Laplacian The discrete Hodge Laplace problem The corresponding discrete Hodge Laplace mixed variational problem is: Find (σ h, u h, p h V h V h H h such that σ h, τ h u h, dτ h =, τ h V h, dσ h, v h + du h, dv h + p h, v h = f, v h, v h V h, u h, q h =, q h H h. ( We can solve the discrete Hodge Laplace problem by first finding the solution (w h, p h V h H h of the Hodge Dirac problem for f, then finding the solution (u h, V h H h of the Hodge Dirac problem for w h. Then (w h du h, u h, p h solves (.

16 Discretization of the Hodge Laplacian Relationship to the discrete Hodge Laplacian Theorem. (LS Under the hypotheses of Theorem., if (σ, u, p V V H solves ( and (σ h, u h, p h V h V h H h solves (, then we have the error estimate σ σ h V + u u h V + p p h ( C inf σ τ V + inf u v V + inf p q V τ V h v V h q V h + µ inf P B u v V. v V h

17 Numerical examples The periodic table of finite elements The elements commonly used in finite element exterior calculus also yield a stable discretization of the Hodge Dirac problem. These elements, including the P r and Pr families of piecewise-polynomial differential forms on simplicial meshes (Arnold Falk and Winther,, and the more recent S r family on cubical meshes (Arnold, ; Arnold and Awanou, give subcomplexes of the L de Rham complex with bounded commuting projections. The P r and Pr families have been implemented in FEniCS and can be used to solve the discrete Hodge Dirac problem. (Arnold, Falk and Winther, ; Arnold ; Arnold and Awanou, ; Logg et al.

18 Hodge Dirac discretization Numerical examples Periodic Table of the Finite Elements k= k= n= P P P RT( RT( r= Nfst ( Ne, tetrahedron, Nfst ( Ne, tetrahedron, ( P, tetrahedron, Nest P Nfnd, ( Ne, tetrahedron, ( P, tetrahedron, DG ( Nf, tetrahedron, ( Q, hexahedron, NfQ DGQ ( NfQ, hexahedron, Symbol of element P ( P, tetrahedron, Dimension of element function space Finite element exterior calculus notation Element specification in FEniCS r= r= k r= n= n= n= n= n= n= n= n= n= 9 n= n= n= n= 9 9 n= n= 9 n= k r= k k The table presents the primary spaces of finite elements for the discretization of the fundamental operators of vector calculus: the gradient, curl, and divergence. A finite element space is a space of piecewise polynomial functions on a domain determined by: ( a mesh of the domain into polyhedral cells called elements, ( a finite dimensional space of polynomial functions on each element called the shape functions, and ( a unisolvent set of functionals on the shape functions of each element called degrees of freedom (DOFs, each DOF being associated to a (generalized face of the element, and specifying a quantity which takes a single value for all elements sharing the face. The element diagrams depict the DOFs and their association to faces. depicted on the left half of the table and The spaces are the two primary families of finite element spaces for meshes of simplices, and the spaces and on the right side are for meshes of cubes or boxes. Each is defined in any dimension n for each value of the polynomial degree r, and each value of k n. The parameter k refers to the operator: the spaces consist of dierential k-forms which belong to the domain of the k th exterior ( S, hexahedron, AAf ( DGS, hexahedron, 9 AAf ( AAe, hexahedron, ( AAe, hexahedron, DGS ( AAf, hexahedron, AAe DGS ( AAf, hexahedron, AAe S ( DGQ, hexahedron, Finite elements Element with degrees of freedom (DOFs DGQ ( NfQ, hexahedron, ( DGS, quadrilateral, ( AAe, hexahedron, ( S, hexahedron, ( DGS, quadrilateral, DGS AAe S ( DGQ, hexahedron, NfQ ( NeQ, hexahedron, ( S, hexahedron, ( BDMS[e,f], quadrilateral, S ( DGQ, hexahedron, ( NeQ, hexahedron, NeQ DGQ ( DGS, quadrilateral, DGS ( BDMS[e,f], quadrilateral, BDMS( ( S, quadrilateral, ( DGQ, quadrilateral, ( NfQ, hexahedron, NeQ Q ( DG, tetrahedron, Legend Weight functions for DOFs S DGS BDMS( ( S, quadrilateral, 9 NfQ ( NeQ, hexahedron, ( Q, hexahedron, ( DGQ, quadrilateral, NeQ Q ( DG, tetrahedron, Nend P ( DG, tetrahedron, DG ( Nf, tetrahedron, DG ( Nf, tetrahedron, Nfnd, ( Ne, tetrahedron, ( P, tetrahedron, DGQ ( Q, hexahedron, ( BDMS[e,f], quadrilateral, S ( RTQ[e,f], quadrilateral, Q ( DG, tetrahedron, Nend P ( DG, tetrahedron, DG ( Nf, tetrahedron, DG ( Nf, tetrahedron, r= Nest ( P, tetrahedron, Nfnd, ( Ne, tetrahedron, ( P, tetrahedron, Nend P ( DG, tetrahedron, ( Q, quadrilateral, ( DG, triangle, DG ( Nf, tetrahedron, P Nfst ( Ne, tetrahedron, ( RTQ[e,f], quadrilateral, RTQ( Q DG ( S, quadrilateral, BDMS( S ( DGQ, quadrilateral, DGQ ( BDM[e,f], triangle, ( P, triangle, Nest ( P, tetrahedron, r= ( DG, triangle, P BDM( ( DGS, interval, DGQ RTQ( ( Q, quadrilateral, ( DG, triangle, P DGS ( S, interval, ( RTQ[e,f], quadrilateral, Q DG ( DGS, interval, S 9 ( BDM[e,f], triangle, ( P, triangle, DG ( RT[e,f], triangle, ( P, triangle, BDM( P ( DG, triangle, ( Q, quadrilateral, DGS RTQ( Q ( DG, triangle, ( S, interval, ( DGQ, interval, DG DG P ( BDM[e,f], triangle, ( P, triangle, ( RT[e,f], triangle, ( P, triangle, r= BDM( ( DGS, interval, S DGQ ( Q, interval, P ( DG, triangle, ( DGQ, interval, k= DGS ( S, interval, DGQ Q ( DG, interval, DG RT( P n= r= ( RT[e,f], triangle, ( P, triangle, k= S DG ( P, interval, k= k= ( DGQ, interval, ( Q, interval, k= DGQ Q ( DG, interval, P ( DG, interval, P DG k= ( Q, interval, ( P, interval, k= k= Q ( DG, interval, P DG ( P, interval, r= k= DG ( DG, interval, k= ( P, interval, DG ( P, interval, r= k= k= P ( DG, interval, r= k= DG ( P, interval, n= k= r= ( DGS, hexahedron, AAf ( AAf, hexahedron, DGS ( DGS, hexahedron, References derivative. Thus for k =, the spaces discretize the Sobolev space H, the domain of the gradient operator; for k =, they discretize H (curl, the domain of the curl; for k = n they discretize H (div, the domain of the divergence; and for k = n, they discretize L. and, which coincide, are the earliest finite The spaces elements, going back in the case r = of linear elements to Courant, and collectively referred to as the Lagrange elements. The and spaces, which also coincide, are the discontinuous Galerkin elements, consisting of piecewise polynomials with no interelement continuity imposed, first introduced by Reed and Hill. The space in dimensions was introduced by Raviart and Thomas and generalized to the -dimensional spaces by Nédélec,while and is due to Brezzi, Douglas and Marini in dimensions, its generalization to dimensions again due to Nédélec. The unified treatment and notation of the and families is due to Arnold, Falk and Winther as part of finite element exterior calculus, extending earlier work of Hiptmair for the family. The space is the span of the elementary forms introduced by Whitney 9. The family of cubical elements can be derived from the -dimensional Lagrange and discontinuous Galerkin elements by a tensor product construction detailed by Arnold, Boffi and Bonizzoni, but for the most part were presented individually along with the corresponding simplicial elements in the papers mentioned. The second cubical family is due to Arnold and Awanou. The finite elements in this table have been implemented as part of the FEniCS Project,,. Each may be referenced in the Unified Form Language (UFL by giving its family, shape, and degree, with the family as shown on the table. For example, the space may be referred to in UFL as: FiniteElement ( Ne, tetrahedron,. Alternatively, the simplicial elements may be accessed in a uniform fashion as: FunctionSpace(mesh, P- Lambda, r, k and for FunctionSpace(mesh, P Lambda, r, k and, respectively. 9 R. Courant, Bulletin of the American Mathematical Society 9, 9. W. H. Reed and T. R. Hill, Los Alamos report LA-UR--9, 9. P. A. Raviart and J. M. Thomas, Lecture Notes in Mathematics, Springer, 9. J. C. Nédélec, Numerische Mathematik, 9. F. Brezzi, J. Douglas Jr., and L. D. Marini, Numerische Mathematik, 9. J. C. Nédélec, Numerische Mathematik, 9. D. N. Arnold, R.S. Falk, and R. Winther, Acta Numerica,. R. Hiptmair, Mathematics of Computation, 999. H. Whitney, Geometric Integration Theory, 9. D. N. Arnold, D. Bo, and F. Bonizzoni, Numerische Mathematik,. D. N. Arnold and G. Awanou, Mathematics of Computation,. A. Logg, K.-A. Mardal, and G. N. Wells (eds., Automated Solution of Differential Equations by the Finite Element Method, Springer,. R. C. Kirby, ACM Transactions on Mathematical Software,. A. Logg and G. N. Wells, ACM Transactions on Mathematical Software,. M. Alnæs, A. Logg, K. B. Øgaard, M. E. Rognes, and G. N. Wells, ACM Transactions on Mathematical Software,. Idea, production and design by Douglas N. Arnold, Anders Logg and Mattias Schläger ( in cooperation with Simula Research Laboratory. This work is licensed under the Creative Commons Attribution ShareAlike License. fenicsproject.org

19 Numerical examples The Hodge Dirac problem allows us to find a vector field with prescribed divergence and curl: Divergence-free vector fields on the unit disk with curl x x. Left: natural boundary conditions (zero normal component. Right: essential boundary conditions (zero tangential component.

20 Further considerations Further considerations On an embedded Riemannian manifold, subdivision into Euclidean simplices introduces geometric errors. Holst and Stern ( have addressed this with their work on geometric variational crimes. This idea has been extended to adaptive mixed methods by Holst, Mihalik and Szypowski ( but the case of Dirac operators on Riemannian manifolds is yet to be tried. Future research could include: numerical examples with more dimensions, more realistic geometries and boundary conditions, perturbed operators, eigenvalue problems, functional calculus, different metric signatures, rougher domains... (Holst and Stern, Holst, Mihalik and Szypowski

21 References Some references D. N. Arnold and G. Awanou, Finite element differential forms on cubical meshes. Mathematics of Computation.,, pp. -. D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numerica,,, pp. -. D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc.,, pp. -. D. N. Arnold, Spaces of finite element differential forms in U. Gianazza et a., (eds. Analysis and Numerics of Partial Differential equations, Springer,, pp. -.

22 References Some references M. Holst and A. Stern, Geometric variational crimes: Hilbert complexes, Finite element exterior calculus, and problems on hypersurfaces. Found. Comput. Math.,, pp. -9. M. Holst, A. Mihalik and R. Szypowski, Convergence and Optimality of Adaptive Mixed Methods on Surfaces. arxiv preprint arxiv:.9,. P. Leopardi and A. Stern. The abstract Hodge-Dirac operator and its stable discretization. arxiv preprint arxiv:.,. A. Logg, K. A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method. Springer,.