Introduction From to Hodge ∗ Discretization Summary

The Discrete

Edmond Rusjan

SUNYIT

HRUMC, April 6, 2013 Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary

Abstract

Exterior Calculus expresses the familiar vector calculus operators , and in a coordinate free notation in terms of the exterior , the wedge and the Hodge star operator. Discrete versions of these operators are important for numerical solutions of partial diﬀerential equations. While the Stokes theorem provides a clear guideline for constructing the discrete , the problem of the discrete Hodge star operator is more subtle. Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Outline

1 Introduction Motivation: Solve PDEs 2 From Vector Calculus to Exterior Calculus Grad, Div, Curl Are All Dead Long Live d, ], [, ∧ and ∗ Example: The 3 Hodge ∗ 2 3 Examples: ∗ in R and R ∗ in Euclidean ∗ on a 4 Discretization From Finite Diﬀerences to Finite Elements From Finite Elements to Discrete Exterior Calculus (DEC) DEC Hodge * FEEC Hodge * 5 Summary Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Motivation: Solve PDEs

Heat equation: u,t = α∆u

Wave: u,tt = ∆u Laplace: ∆u = 0 and Poisson : ∆u = f ﬂow: k Darcy: v + µ ∇p = 0, ∇ · v = 0 ∂v ∇p Euler: ∂t + v · ∇v = − ρ + g ∂v Navier-Stokes: ρ( ∂t + v · ∇v) = −∇p + ∇ · T + f :

Euler-Bernoulli Beam: ρAu,tt + [EIuxx ]xx = p(x, t) 4 Kirchhoﬀ Plate: ρhu,tt + D∇ u = q(x, y, t) - Maxwell: ˙ ˙ ∇ · B~ = 0, ∇ · E~ = ρ, ∇ × E~ = −B~ , ∇ × B~ =~j + E~ 2 ~ Quantum - Schroedinger: − 2m ∆Ψ + V Ψ = i~Ψt Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Gradient: steepest ascent, heat seeking bug

If f (x, y) is the height of the hill at x, y, then ∇f is the steepest ascent direction (locally) Let f (t, x, y, z) be temperature at time t and x, y, z Gradient ∇f points in the direction of the fastest temperature increase k∇f k tells how fast is the fastest increase ∂f ∂ ∂f ∂ ∂f ∂ in Cartesian coordinates ∇f = ∂x ∂x + ∂y ∂y + ∂z ∂z     f,x 1 ∂f ∂ ∇f =  f,y  , f,x = ∂x , ∂x =  0  = ∂x f,z 0 Heat seeking bug travels in the ∇f direction Diﬀerential of f - the best linear approximation to f df : Rn → R is a linear df (v) = ∇f · v - , if kvk = 1 df = f,x dx + f,y dy + f,z dz = [f,x , f,y , f,z ], dx = [1, 0, 0] df = [∇f ]T {dx, dy, dz} is the dual to {∂x , ∂y , ∂z } Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary between ∇f and df

Two dual points of view: ∇f and df More precisely: 3 3 At a ﬁxed point p ∈ M = R : ∇f |p ∈ V = TpM = R 0 ∗ 3 0 Dual point of view: df |p ∈ V = Tp M = (R ) As the point p ∈ M varies: ∇f ∈ TM, the Dual point of view: df ∈ T ∗M, the The V and its dual V 0 are isomorphic: natural between V and V 0 0 [ ﬂat: [ : V → V , ∂x = dx, etc. 0 [ sharp: ] : V → V , dx = ∂x , etc. extend by linearity more generally, ﬂat: [ : V → V 0, ~u[(~v) = ~u · ~v sharp: ] : V 0 → V , ] = [−1 Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary ∇f and df compete on a manifold M: df wins

In Cartesian coordinates in , it’s a tie ∇f and df equally convenient because the length element is ds = pdx 2 + dy 2 Consider polar coordinates: x = r cos θ, y = r sin θ.

df is coordinate independent df = f,r dr + f,θdθ ∇f is not coordinate independent ∇f 6= f,r ∂r + f,θ∂θ length element is not ds2 = dr 2 + dθ2 dx = x,r dr + x,θdθ = cos θdr − r sin θdθ dy = y,r dr + y,θdθ = sin θdr + r cos θdθ ds2 = dx 2 + dy 2 = dr 2 + r 2dθ2 Introduce g to measure 2 P i j ds = ij gij dx dx . 2 i j Einstein convention ds = gij dx dx .  1 0  g = , ij 0 r 2 Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Grad is Dead

Metric tensor g and its inverse g −1     1 0 ij 1 0 gij = 2 , g = 1 , 0 r 0 r 2 Generalize ﬂat: [ : TM → T ∗M from ~u[(~v) = ~u · ~v to ~u[(~v) = g(~u,~v) In components: k Vector ~u = u ∂k i j i i g = gij dx dx , dx (∂j ) = δj Co-vector or 1-form ~u[ = g(u, ·) [ i j i k j k i j i j ~u = gij dx (u)dx = gij dx (u ∂k )dx = gij u δk dx = gij u dx [ j Standard practice to write ~u = uj dx i By comparison uj = gij u −1 i ij ] = [ , so u = g uj ∇f = (df )] i ij (∇f ) = g f,j r rj rr (∇f ) = g f,j = g f,r + 0 = f,r θ θj θθ 1 (∇f ) = g f,j = 0 + g f,θ = r f,θ Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Div and Curl are Dead

] r θ 1 Grad ∇f = (df ) = (∇f ) ∂r + (∇) ∂θ = f,r ∂r + r ∂θ Curl ∇ × F = [∗(dF [)]] This deﬁnes the exterior derivative d Exterior, because takes 1-form into a 2-form Anti-symmetric, because anti-symmetric Need a symbol to denote 2-forms: ∧ wedge product Extend by linearity and associativity: exterior This introduces the Hodge ∗ Div ∇ · F = ∗d(∗F [) Laplace ∇2f = ∗d(∗df ) Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary My Cup of Tea

Consider a cup of hot tea, which is cooling down u(x, y, x, t) - temperature at x, y, z at time t Heat ﬂows from the tea into the surrounding from hot to cold, so in the −∇u direction Fourier’s Law of thermal conduction: ~q = −k∇u ~q the heat ﬂux density k Divergence tells how fast heat is ﬂowing out: ∇ · ~q Energy (Heat) Conservation: tea cools down because heat ﬂows out

(ρcpu)t = −∇ · ~q = −∇ · (−k∇u) ρ density cp thermal capacity at constant Heat Equation: u = α∇2u, α = k t ρcp Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Stokes Theorem

Consider a body B of tea, not necessarily all the tea If B is warmer than the surroundings, heat ﬂows through the boundary ∂B from B into the surroundings BC : H ~ Φ = ∂B ~q · dS The rate of change of thermal energy (heat) inside a body B equals the heat ﬂux into B minus the heat ﬂux out of B through the boundary ∂B Let B be inﬁnitesimal, with volume dV

Heat inside B: E = ρcpudV Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Hodge ∗ Example

2 V = R : ∗1 = e1 ∧ e2 ∗e1 = e2 ∗e2 = −e1 ∗(e1 ∧ e2) = 1 3 V = R : ∗1 = e1 ∧ e2 ∧ e3 ∗e1 = e2 ∧ e3 ... = ... ∗(e1 ∧ e2) = e3 ... = ... ∗(e1 ∧ e2 ∧ e3) = 1 Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Hodge ∗

V oriented : ∗ : Vp V → V(n−p) V natural isomorphism

Let {e1, e2,..., en} be a basis for V

Let σ = e1 ∧ e2 ∧ · · · ∧ en be a chosen Hodge ∗: u ∧ v = (∗u, v)σ Intuition: complementary n − p vector to the given p vector orthogonal consistent with the orientation Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary ∗ on a Manifold

p ∗ J ∗α = αJ dx ∗ p K αJ = |g|α KJ p In full: α∗ = |g| P αk1,...,kp  j1,...,jn−p k1<···

Finite Diﬀerences: time permitting, work on board Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Finite Elements

Finite Elements: time permitting, work on board Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Discrete d

d is uniquely determined by the Stokes theorem: the of the discrete boundary operator Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary MDEC : DEC Hodge *

1 R 1 R Marsden, Hirani, Desbrun, et al.: |∗σ| ∗σ ∗α = |σ| σ α k DEC |∗σi | [M ]ij = k δij |σi | Advantage: MDEC diagonal Disadvantage: MDEC not positive deﬁnite, because circumcenter may be outside the Example: standard 2-simplex; vertices: (0, 0), (1, 0), (0, 1)

 1  2 0 0 DEC 1 M1 =  0 2 0  0 0 0 Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary MWhit: FEEC Hodge *

Whitney 0-forms are barycentric coordinates:

Whitney 1-forms: ηk = λi dλj − λj dλi Whit R [M ]ij = ηi ηj dA Advantage: MWhit is positive deﬁnite, because barycenter is always inside the simplex Disadvantage: MWhit not diagonal, (MWhit )−1 not sparse Example: standard 2-simplex; vertices: (0, 0), (1, 0), (0, 1)

 1 1  3 6 0 Whit 1 1 M1 =  6 3 0  1 0 0 6 Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Summary

Exterior Calculus is a coordinate independent language and consequently superior in and on curved surfaces. Discrete Exterior calculus has the potential to revolutionize the numerical PDE ﬁeld. The discrete exterior derivative d is unique and determined by the discrete Stokes theorem. The discrete Hodge * operator is an open question and a topic of current research. Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary References

Abraham, Marsden, Ratiu, , Tensor Analysis and Applications. Bell, Hirani, PyDEC: Software and for Discretization of Exterior Calculus. Desbrun, Kanso, Tong, Discrete Diﬀerential Forms. Flanders, Diﬀerential Forms with Applications to the Physical Sciences. Frankel, The Geometry of . Gillette, Notes on Discrete Exterior Calculus. Hirani, Nakshatrala, Chaudhry, Numerical method for Darcy ﬂow derived using Discrete Exterior Calculus. Hirani, Kalyanaraman, Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods. Lee, Riemannian Manifolds. Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Thank You!

THANK YOU!!!