Motivations. Outline. Finite element exterior calculus and the geometrical basis of numerical stability. References. Douglas N.

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1 Outline Finite element exterior calculus and the geometrical basis of numerical stability Douglas N. Arnold joint with R. Falk and R. Winther School of Mathematics University of Minnesota April 009 Motivations Exterior calculus and PDE 3 Discretization 4 Finite element differential forms 5 Application to elasticity / 40 / 40 References Finite element exterior calculus, homological techniques, and applications, Acta Numerica 006, p. 55 Any young (or not so young) mathematician who spends the time to master this paper will have tools that will be useful for his or her entire career. Math Reviews Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp. 007 Motivations Why do we need more theory for finite elements? Geometric decompositions and local bases for spaces finite element differential forms, to appear in CMAME everything is at arnold 3 / 40

2 Why do we need FEEC? The finite element method is incredibly successful. FEM often amenable to mathematical analysis, allowing validation and comparison of methods. But plenty of challenges remain, for algorithms and analysis! Approximability, consistency, and stability = convergence Stability, like well-posedness, can be extremely subtle Well-posedness + approximability + consistency stability Exterior calculus, de Rham cohomology, Hodge theory,..., are geometric tools to get at well-posedness. FEEC adapts these tools to the discrete level to get at stability. 5 / 40 Steady heat conduction problem: finite elements in H div C grad u = f strong C grad u grad v dx = f v dx v weak ( ) C grad u grad u f u u dx minimum variational grad u dx < u H () H : u L (), grad u L (; R n ) The right FE space: Lagrange elements { v H () : v T Pr (T ) T Th } 6 / 40 Lagrange finite elements First order (mixed) formulation Shape fns: P DOFs: vertex vals. & edge averages A σ = grad u, divσ = f strong A σ τ dx = div τ u dx τ, div σ v dx = f v dx v ( A σ σ + div σ u + f u) dx σ, u stationary pt. weak variational P P P3 σ H(div, ), u L () Lagrange elements? Unstable! 7 / 40 8 / 40

3 Thermal problem in D σ = u, σ = f on (, ) (σ + σ σ, u u + f u) dx stationary point H L Babuška Narasimhan Thermal problem in D σ = grad u, div σ = f ( σ σ, u + div σ u + f u) dx stationary point H(div) L P P0 P Λ P0 P-P (0 elts) P-P (40 elts) P-P0 (40 elts) 9 / 40 0 / 40 Raviart Thomas elements (P r Λ ) Shape functions P Λ : span [ ( 0), ( 0 ), ( x y) ] DOFs: A mixed FEM for nd order elliptic problems, Proc. conf. Math l Aspects of the FEM, Rome 975. Springer Lect. Notes in Math #606, 977. Generalizes to all degrees, and all dimensions (n = 3: Nédélec 80) Math & CS SIAM J. Numerical Analysis Numerische Mathematik Mathematics of Computation RAIRO M AN Num. Methods for PDEs Eng. & Apps CMAME Computational Geosciences J. Computational Physics IJNME COMPEL Maxwell eigenvalue problem, unstructured mesh µ curl E curl Ẽ = ω ɛe Ẽ Ẽ Right space is H(curl) λ = m + n = 0,,,, 4, 4, 5, 5, 8, (Lag.P) P Λ / 40 / 40

4 Maxwell eigenvalue problem, regular mesh Vector Laplacian curl curl u grad div u = f in Ω λ = m + n =,,, 4, 4, 5, 5, 8,... u n = 0, Z Ω rot u = 0 on Ω u ( curl u + div u ) f u minimum Lagrange finite elements converge nicely but not to the solution! (same problem with any conforming FE) A mixed formulation based on appropriate finite elements works fine Z Boffi-Brezzi-Gastaldi 99 3 / 40 Ω σ,u ( σ curl σ u div u f u) dx stationary point H(curl) H(div) 4 / 40 EM calculations based on the generalized RT elements Scho berl, Zaglmayr 006, NGSolve Exterior calculus and PDE The continuous problem K tets, P6 Λ Also: White EMSolve, Demkowicz 3Dhp90, Durufle Montjoie,... 6K tets, P3 Λ 5 / 40

5 Differential forms For R n, Λ k () consists of functions Alt k R n so if ω Λ k (), ωx(v,..., vk) R, x, vi R n Λ 0 (): real-valued functions on Λ (): covector fields, ω = n i= fidxi, fi functions (dxi(ej) = δij) Λ (): ω = i<j fijdxi dxj, (dxi dxj := dxi dxj dxj dxi) De Rham complex and cohomology 0 Λ 0 () d0 Λ () d dn Λ n () 0 # of components, i = 0, Zk := ker(d k ) # of holes, i =, Bk := range(d k dim Zk/Bk = ) # of voids, i =, vector proxies in R 3 : fidxi (f, f, f3), fijdxidxj (f3, f3, f) 0 C () grad C (; R 3 ) curl C (; R 3 ) div C () 0 d k : Λ k () Λ k+ () d(f dxj dxk) = i ω R ω Λ k, dim f = k f f xi dxi dxj dxk Physical vector quantities may be divided into two classes, in one of which the quantity is dened with reference to a line, while in the other the quantity is dened with reference to an area. James Clerk Maxwell, Treatise on Electricity & Magnetism, 89 7 / 40 8 / 40 Hodge theory Making use of the inner product: Hodge star: : Λ k () = Λ n k () formal adjoint: δ = ± d : Λ k () Λ k () Λ k () d δ Λ k () d Λ k+ () δ Hodge Laplacian: dδ + δd : Λ k Λ k harmonic forms: H k := { ζ Z k ζ B k } = Z k /B k Hodge decomposition: L Λ k () = B k H k (Z k ) Poincaré s inequality: ω L c dω L, ω (Z k ) Sobolev spaces: HΛ k () = { ω L Λ k () dω L Λ k+ () } 0 HΛ 0 () 0 H () d HΛ () grad H(curl, ) d curl H(div, ) d HΛ n () 0 div L () 0 Some applications Physical quantities: 0-forms: temperature; electric field potential -forms: temperature gradient; electric field -forms: heat flux; magnetic flux stress is a covector-valued -form 3-forms: heat density; charge density; mass density PDEs: div grad u = f (curl curl grad div)u = f curl curl u = f, div u = 0 div u = f, curl u = 0 Maxwell s equations elasticity dynamic problems, eigenvalue problems, lower order-terms variable coefficients, nonlinearities... 9 / 40 0 / 40

6 Hodge Laplace problem Well-posedness of the Hodge Laplacian Given f Λ k (0 k n), find u Λ k with (dδ + δd)u = f (plus BC) Harmonic functions determine well-posedness: u f H k, u is determined only mod H k This mixed formulation is always well-posed: Given f L Λ k (), find σ HΛ k, u HΛ k, p H k : σ, τ dτ, u = 0 τ HΛ k dσ, v + du, dv + p, v = f, v v HΛ k u, q = 0 q H k Equivalently σ, σ du, du dσ, u u, p + f, u saddle point / 40 σ HΛ k, u HΛ k, p H k : σ, τ dτ, u = 0 dσ, v + du, dv + p, v = f, v u, q = 0 τ HΛ k v HΛ k q H k Need to control σ HΛ + u HΛ + p by a bounded choice of τ, v, and q. τ = σ controls σ, v = dσ controls dσ, v = p controls p v = u controls du, How to control u?? Hodge decomp.: u = dη + s + z, η HΛ k, s H k, z (Z k ) τ = η controls dη and q = s controls s. To bound z we use Poincaré s inequality: z c dz = c du (which is under control) / 40 Abstract setting Λ k d k Λ k Λ k+ d k Discretization We have a well-posed variational PDE problem. How do we discretize it stably? Λ k h d k Λ k h d k Λ k+ h Complex of Hilbert spaces with d k bounded and closed range. For discretization, construct a finite dimensional subcomplex. Define H k h = (Bk h ) Z k h. Discrete Hodge decomp. follows: Λ k h = Bk h Hk h (Zk h ) Galerkin s method: Λ k, Λ k, H k Λ k h, Λ k h, Hk h When is it stable? 4 / 40

7 Bounded cochain projections Key property: The finite dimensional subcomplex admits a bounded cochain projection. Λ k d k Λ k π k h Λ k h π k h d k Λ k h π k h bounded π k h a projection πh kd k = d k π k h limh 0 πh k v = v, v Λk Theorem The induced map on cohomology is an isomorphism for h small. gap ( H k, H k h) 0 The discrete Poincaré inequality holds uniformly in h. Galerkin s method is stable and convergent. Proof of discrete Poincaré inequality Thm. There is a positive constant c, independent of h, such that ω c dω, ω Z k h. Proof. Given ω Z k h, define η Zk HΛ k () by dη = dω. By the Poincaré inequality, η c dω, so it is enough to show that ω c η. Now, ω πhη Λ k h and d(ω πhη) = 0, so ω πhη Z k h. Therefore ω = ω, πhη + ω, ω πhη = ω, πhη ω πhη, whence ω πhη, and the result follows. 5 / 40 6 / 40 Finite element differential forms Finite element differential forms How do we construct finite element spaces that fit together in de Rham subcomplexes with bounded cochain projections? Let T = Th be a triangulation of R n. We wish to construct finite element spaces Λ k (T ) HΛ k () which form a finite dimensional subcomplex with bounded cochain projections. We will construct them as usual for finite elements: On each simplex T T we specify a space of polynomials shape functions degrees of freedom, each associated to a face of the simplex It turns out that for each form degree k and polynomial degree r, there are just two natural finite element subspaces of HΛ k (): Pr Λ k (Th) and P r Λ k (Th) Pr Λ k (Th), k = 0, Pr Λ k (Th) = Pr Λ k (Th), k = n, strictly between, 0 < k < n 8 / 40

8 The Koszul complex Key tool: the Koszul differential κ : Λ k Λ k : Definition of P r Λ k (κω)x(v,..., v k ) = ωx(x, v,..., v k ), X = x x0 0 Pr Λ 0 κ Pr Λ κ κ Pr nλ n 0 C.f., the polynomial de Rham complex 0 Pr Λ 0 d Pr Λ d d Pr nλ n 0 For R 3 0 Pr () X Pr (; R 3 ) Key relation: (dκ + κd)ω = (r + k)ω X Pr (; R 3 ) X Pr 3() 0 ω Hr Λ k (homogeneous polys) Pr Λ k = Pr Λ k + Hr Λ k = Pr Λ k + κhr Λ k+ + dhr+λ k P r Λ k := Pr Λ k + κhr Λ k+ God made P rλ k and P r Λ k, all the rest is the work of man. Hr Λ k = dhr+λ k κhr Λ k+ 9 / / 40 Degrees of freedom Finite element differential forms and classical mixed FEM The other ingredient of a finite element space are the degrees of freedom, i.e., a decomposition of the dual spaces (Pr Λ k (T )) and (P r Λ k (T )), into subspaces associated to subsimplices f of T. DOF for Pr Λ k (T ): to a subsimplex f of dim. d k we associate ω Trf ω η, η P r+k d Λd k (f ) f DOF for Pr Λ k (T ): ω Trf ω η, η Pr+k d Λ d k (f ) Hiptmair f The resulting FE spaces have exactly the continuity required by HΛ k : Theorem. Pr Λ k (T ) = { ω HΛ k () : ω T Pr Λ k (T ) T T }. Similarly for P r. 3 / 40 Pr Λ 0 (T ) = Pr Λ 0 (T ) H Lagrange elts Pr Λ n (T ) = Pr Λ n (T ) L discontinuous elts n = : Pr Λ (T ) H(curl) Raviart Thomas elts n = : P r Λ (T ) H(curl) Brezzi Douglas Marini elts n = 3: Pr Λ (T ) H(curl) Nedelec st kind edge elts n = 3: P r Λ (T ) H(curl) Nedelec nd kind edge elts n = 3: Pr Λ (T ) H(div) Nedelec st kind face elts n = 3: P r Λ (T ) H(div) Nedelec nd kind face elts 3 / 40

9 Finite element de Rham subcomplexes From these spaces we want to build discrete de Rham complexes with bounded projections. It turns out that there are lots of ways to do this ( n for each r). Extreme cases are: 0 P r Λ 0 (T ) d P r Λ (T ) d d Pr Λ n (T ) 0 0 grad Whitney 957, Bossavit 988 curl div 0 Application to Elasticity What else can you do with FEEC? 0 Pr Λ 0 (T ) d Pr Λ (T ) d d Pr nλ n (T ) 0 0 grad curl div 0 33 / 40 Stress displacement mixed finite elements for elasticity Find stress σ : R 3 3 sym, displacement u : R 3 such that Aσ = ɛ(u), div σ = f ( ) σ,u A σ :σ + div σ u + f u dx stationary point H(div;S) L (R n ) Recent progress coming from the FEEC perspective First stable elements based on polynomials, D (Arnold Winther 00), all degrees r : Search for stable finite elements dates back to the 60s, very limited success.... to derive elements that exhibit complete continuity of the appropriate components along interfaces... was achieved by Raviart and Thomas in the case of the heat conduction problem.... Extension to the full stress problem is difficult and as yet such elements have not been successfully noted. Zienkiewicz, Taylor, Zhu The Finite Element Method: Its Basis & Fundamentals, 6th ed., vol., 005 3D stable elements, all degrees r (Arnold Awanou Winther 007): for r = stress space has 6 degrees of freedom (7 per component on average) 35 / / 40

10 A computation using the new elements From Eberhard, Hueber, Jiang, Wohlmuth 006 Mixed formulation with weak symmetry Idea goes back to Fraeijs de Veubeke 975, Amara Thomas 979 In the classical Hellinger Reissner principle, symmetry of the stress tensor (balance of angular momentum) is assumed to hold exactly. Instead we impose it weakly with a Lagrange multiplier (the rotation). ( ) σ,u A σ :σ + div σ u + f u dx stationary point H(div;S) L (R n ) ( ) σ,u,p A σ :σ + div σ u + σ :p + f u dx S.P. H(div;M) L (R n ) L (K) FEEC has led to very simple stable elements 37 / 40 σ u p 38 / 40 Features of the new mixed elements Based on HR formulation with weak symmetry; very natural Lowest degree element is very simple: full P for stress, P0 for displacement and rotation Works for every polynomial degree Works the same in and 3 (or more) dimensions Robust to material constraints like incompressibility Provably stable and convergent 39 / 40 Conclusions Exterior calculus clarifies the nature of physical quantities and the structure of the PDEs involving them. Capturing the right structure on the discrete level can be essential to get stable methods. FEEC provides a very natural framework for the design and understanding of subtle stability issues that arise in the discretization of a wide variety of PDE systems. It brings to bear tools from geometry, topology, and algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structure of the PDE system, and so obtain stability. FEEC has been used to unify, clarify, and refine many known finite element methods. It is a mathematically rigorous theory. The Pr Λ k and P r Λ k spaces are the natural finite element discretizations for differential forms and the de Rham complex. Through FEEC we believe we have completed the long search for just the right mixed finite elements for elasticity. 40 / 40