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The Discrete Hodge Star Operator Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary The Discrete Hodge Star Operator 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 gradient, divergence and curl in a coordinate free notation in terms of the exterior derivative, the wedge product 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 exterior derivative, 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 Heat Equation 3 Hodge ∗ 2 3 Examples: ∗ in R and R ∗ in Euclidean Space ∗ on a Manifold 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 Fluid flow: k Darcy: v + µ rp = 0; r · v = 0 @v rp Euler: @t + v · rv = − ρ + g @v Navier-Stokes: ρ( @t + v · rv) = −∇p + r · T + f Elasticity: Euler-Bernoulli Beam: ρAu;tt + [EIuxx ]xx = p(x; t) 4 Normal Kirchhoff Plate: ρhu;tt + Dr u = q(x; y; t) Electromagnetism - Maxwell: _ _ r · B~ = 0; r · E~ = ρ, r × E~ = −B~ ; r × B~ =~j + E~ 2 ~ Quantum mechanics - 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 rf is the steepest ascent direction (locally) Let f (t; x; y; z) be temperature at time t and position x; y; z Gradient rf points in the direction of the fastest temperature increase krf k tells how fast is the fastest increase @f @ @f @ @f @ in Cartesian coordinates rf = @x @x + @y @y + @z @z 2 3 2 3 f;x 1 @f @ rf = 4 f;y 5 ; f;x = @x ; @x = 4 0 5 = @x f;z 0 Heat seeking bug travels in the rf direction Differential of f - the best linear approximation to f df : Rn ! R is a linear functional df (v) = rf · v - directional derivative, if kvk = 1 df = f;x dx + f;y dy + f;z dz = [f;x ; f;y ; f;z ]; dx = [1; 0; 0] df = [rf ]T fdx; dy; dzg is the dual basis to f@x ;@y ;@z g Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Duality between rf and df Two dual points of view: rf and df More precisely: 3 3 At a fixed point p 2 M = R : rf jp 2 V = TpM = R 0 ∗ 3 0 Dual point of view: df jp 2 V = Tp M = (R ) As the point p 2 M varies: rf 2 TM, the tangent bundle Dual point of view: df 2 T ∗M, the cotangent bundle The vector space V and its dual V 0 are isomorphic: natural isomorphism 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 rf and df compete on a manifold M: df wins In Cartesian coordinates in Euclidean space, it's a tie rf 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 rf is not coordinate independent rf 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 metric tensor g to measure distances 2 P i j ds = ij gij dx dx . 2 i j Einstein summation 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 Metric tensor 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 rf = (df )] i ij (rf ) = g f;j r rj rr (rf ) = g f;j = g f;r + 0 = f;r θ θj θθ 1 (rf ) = 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 rf = (df ) = (rf ) @r + (r) @θ = f;r @r + r @θ Curl r × F = [∗(dF [)]] This defines the exterior derivative d Exterior, because takes 1-form into a 2-form Anti-symmetric, because cross product anti-symmetric Need a symbol to denote 2-forms: ^ wedge product Extend by linearity and associativity: exterior algebra This introduces the Hodge ∗ Div r · F = ∗d(∗F [) Laplace r2f = ∗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 volume from hot to cold, so in the −∇u direction Fourier's Law of thermal conduction: ~q = −kru ~q the heat flux density k thermal conductivity Divergence tells how fast heat is flowing out: r · ~q Energy (Heat) Conservation: tea cools down because heat flows out (ρcpu)t = −∇ · ~q = −∇ · (−kru) ρ density cp thermal capacity at constant pressure Heat Equation: u = αr2u; 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 inner product space: ∗ : Vp V ! V(n−p) V natural isomorphism Let fe1; e2;:::; eng be a basis for V Let σ = e1 ^ e2 ^ · · · ^ en be a chosen orientation 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 = jgjα KJ p In full: α∗ = jgj P αk1;:::;kp j1;:::;jn−p k1<···<kp k1;:::;kp;j1;:::;jn−p Introduction From Vector Calculus to Exterior Calculus Hodge ∗ Discretization Summary Finite Differences 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 transpose 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.: |∗σj ∗σ ∗α = jσj σ α k DEC |∗σi j [M ]ij = k δij jσi j Advantage: MDEC diagonal Disadvantage: MDEC not positive definite, because circumcenter may be outside the simplex Example: standard 2-simplex; vertices: (0; 0), (1; 0), (0; 1) 2 1 3 2 0 0 DEC 1 M1 = 4 0 2 0 5 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) 2 1 1 3 3 6 0 Whit 1 1 M1 = 4 6 3 0 5 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 curvilinear coordinates and on curved surfaces. Discrete Exterior calculus has the potential to revolutionize the numerical PDE field.
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