Finite Element Clifford Algebra: A New Toolkit for Evolution Problems

Transcription

1 Finite Element Clifford Algebra: A New Toolkit for Evolution Problems Andrew Gillette joint work with Michael Holst Department of Mathematics University of California, San Diego agillette/ Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

2 Motivation Poisson s equation: Given f find u(x) such that { 0 = u + f in Ω R n u = 0 on Ω a Heat equation: Given f and g, find u(x, t) such that u t = u + f in Ω R n, for t > 0, u = 0 on Ω, for t > 0, u t=0 = g in Ω Finite element exterior calculus (FEEC) provides: abstract framework for analyzing numerical approximation of elliptic PDEs classification of stable finite element methods with optimal convergence rates How can the FEEC framework be expanded to classify stable finite element methods for evolutionary PDEs? Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

3 Outline of approach Two possible methods for extending Finite Element Exterior Calculus: Semi-discrete: Finite element method in space, ODE in time t Ω domain: Ω [0, T ] R n R solution basis: φ h t=t0 : Ω R, for each t 0 [0, T ] error analysis: FEEC + Bochner space theory Fully discrete: Finite element method in space and time Ω domain: solution basis: error analysis: Ω [0, T ] R n R φ h : Ω [0, T ] R Finite Element Clifford Algebra This talk: Initial results on semi-discrete approach + a preview of FECA Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

4 Table of Contents 1 Motivation 2 Background: FEEC, Bochner Spaces, Semi-Discrete methods 3 Results: New error estimates in Bochner norms 4 Preview: Why FECA is needed Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

5 Outline 1 Motivation 2 Background: FEEC, Bochner Spaces, Semi-Discrete methods 3 Results: New error estimates in Bochner norms 4 Preview: Why FECA is needed Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

6 Finite Element Exterior Calculus in context Consider a mixed method for Poisson s problem on a domain Ω R n : continuous u + f = 0, u H 2 mixed weak (div σ, φ) + (f, φ) = 0, φ L 2 = Λ n (σ, ω) + (u, div ω) = 0, ω H(div) = Λ n 1 mixed FEM (div σ h, φ h ) + (f, φ h ) = 0, φ h Λ n h L 2 (σ h, ω h ) + (u h, div ω h ) = 0, ω h Λ n 1 h H(div) Major Conclusions from FEEC The finite elements spaces Λ n 1 h and Λ n 1 h should be chosen from two classes of piecewise polynomial spaces, denoted P r Λ k h and Pr If this choice is made in a compatible manner implied by the exterior calculus structure, then optimal a priori error estimates are guaranteed Λ k h Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

7 Finite Element Exterior Calculus in context Theorem [Arnold, Falk, Winther; Bulletin of AMS, 2010] Assume the elliptic regularity estimate u H s+2 + u H s+1 + σ H s+1 + div σ H s c f H s holds for 0 s s max. Choose finite element spaces Λ n 1 h =, Λn h = P r+1 Λ n 1 (T ) or P r+1 Λn 1 (T ) P r+1 Λn (T ) or P r Λ n (T ) Then for 0 s s max, the following error estimates hold { ch f L 2 if Λ n h = P 1 u u h L 2 Λn (T ), ch 2+s f H s otherwise, if s r 1 σ h σ L 2 ch s+1 f H s if { Λ n 1 h = P r+1 Λ n 1, s r + 1, Λ n 1 h = P r+1 Λn 1, s r, div (σ h σ) L 2 ch s f H s, if s r + 1. Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

8 Semi-discrete Mixed Formulation Consider a mixed method for the heat equation on Ω R n for t I := [0, T ]. continuous u t u = f, u t=0 = g. mixed weak (u t, φ) (div σ, φ) = (f, φ), φ Λ n, t I, (σ, ω) + (u, div ω) = 0, ω Λ n 1, t I, u t=0 = g. mixed FEM (u h,t, φ h ) (div σ h, φ h ) = (f, φ h ), φ h Λ n h, t I, (σ h, ω h ) + (u h, div ω h ) = 0, ω h Λ n 1 h, t I, u h t=0 = g h. linear system AU t BΣ = F B T U + DΣ = 0 AU t + BD 1 B T U = F Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

9 Semi-discrete Error Bounds Theorem [Thomée; Galerkin FEM for Parabolic Problems, 1997] Fix n = 2 and set Λ 2 h := discontinuous linear, Λ 1 h := Raviart-Thomas elements. Let g h be the solution to the elliptic problem with f = g. Then for t 0: t ) u h (t) u(t) L 2 ch ( u(t) 2 H 2 + u t H 2 ds, 0 ( t ) 1/2 ) σ h (t) σ(t) L 2 ch ( u(t) 2 H 3 + u t 2 H ds. 2 0 Homogeneous case (f = 0), g h as above, t 0: u h (t) u(t) L 2 ch 2 g H 2, if g Ḣ2, σ h (t) σ(t) L 2 ch 3 g H 3, if g Ḣ3. Homogeneous case (f = 0), g h := orthogonal projection of g on to Λ 2 h, t > 0: u h (t) u(t) L 2 ch 2 t 1 g L 2 σ h (t) σ(t) L 2 ch 2 t 3/2 g L 2 Note: These bounds are space-only and restricted to the case n = 2. Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

10 Bochner spaces and norms Our new error bounds will employ the theory of Bochner spaces Definition Let X be a Banach space and I = (0, T ). Define Equip this space with the norm C(I, X) := {u : I X u bounded and continuous} u C(I,X) := sup u(t) X. t I The Bochner space L P (I, X) is defined to be the completion of C(I, X) with respect to the norm: ( ) 1/p u L p (I,X) := u(t) p X dt. We combine notations to get Bochner differential form spaces: I L 2 X k := L 2 (I, L 2 Λ k (Ω)) These are parametrized differential form spaces. Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

11 Outline 1 Motivation 2 Background: FEEC, Bochner Spaces, Semi-Discrete methods 3 Results: New error estimates in Bochner norms 4 Preview: Why FECA is needed Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

12 Bochner-FEEC Parabolic Error Estimates We combine the FEEC and parabolic error estimates to derive the following. Theorem [G, Holst, 2011] Let n 2 and fix I := [0, T ]. Suppose regularity estimate u(t) H s+2 + u(t) H s+1 + σ(t) H s+1 + div σ(t) H s c f (t) H s holds for 0 s s max and t I. Choose finite element spaces P r+1 Λ n 1 (T ) P Λ n 1 h = or P, r+1 Λn (T ) Λn h = or r+1 Λn 1 (T ) P r Λ n (T ) Then for 0 s s max and g h the solution to the elliptic problem we have ( ch f L 2 (I,L 2 ) + ) T f t L 1 (I,L 2 ) if Λ n h = P 1 Λn (T ) u h u L 2 X n ( ch 2+s f L 2 (I,H s ) + ) T f t L 1 (I,H s ) otherwise, if s r 1 and... Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

13 Bochner-FEEC Paraoblic Error Estimates Theorem [G, Holst, 2011] Let n 2 and fix I := [0, T ]. Suppose regularity estimate u(t) H s+2 + u(t) H s+1 + σ(t) H s+1 + div σ(t) H s c f (t) H s holds for 0 s s max and t I. Choose finite element spaces P r+1 Λ n 1 (T ) P Λ n 1 h = or P, r+1 Λn (T ) Λn h = or r+1 Λn 1 (T ) P r Λ n (T ) Then for 0 s s max and g h the solution to the elliptic problem we have: Λ n 1 h = P 1 Λ n 1 (T ), s 1 If or Λ n 1 h = P and Λn h = P 1 Λn (T ), then 1 Λn 1 (T ), s = 0 ( ) σ h σ L 2 X c h 1+s f n 1 L 2 (I,H s ) + h T f t L 2 (I,L 2 ) For any other choice of spaces, if s r 1, ( ) σ h σ L 2 X c h 1+s f n 1 L 2 (I,H s ) + h2+s T f t L 2 (I,L 2 ) Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

14 Proof and Significance Key idea of the proof: u(t) u h (t) L 2 }{{} error between weak and semi-discrete u(t) ũ h (t) L 2 }{{} error between weak and time-ignorant elliptic + ũ h (t) u h (t) L 2 }{{} error between time-ignorant elliptic and semi-discrete Significance of the error estimates These results give a priori estimates of convergence rates for the semi-discrete Galerkin FEM for the heat equation. By using the FEEC framework, we have classified choices of semi-discrete finite element spaces that guarantee optimal convergence rates. The results hold for arbitrary spatial dimension n, not just n = 2. For the homogeneous case (f = 0) with sufficiently regular g, we expect to find stronger error estimates akin to Thomée s. G, HOLST, Finite Element Exterior Calculus for Evolution Problems, in preparation. Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

15 Outline 1 Motivation 2 Background: FEEC, Bochner Spaces, Semi-Discrete methods 3 Results: New error estimates in Bochner norms 4 Preview: Why FECA is needed Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

16 The Bochner Complex FEEC theory studies discretizations of the L 2 derham complex: 0 HΛ 0 d Ω (grad) HΛ 1 d Ω d Ω (div) HΛ n d Ω 0 We can define a parametrized exterior derivative operator on Bochner spaces: d : HX k HX k+1 where (dµ)(t) := d Ω (µ(t)). This gives rise to a Bochner domain complex: 0 HX 0 d HX 1 d d HX n d 0 Ω For a fully discrete method, we need an exterior derivative operator on spacetime elements which can distinguish spacelike and timelike dimensions. Such an operator needs the Lorentzian signature of basis elements - a tool available in Clifford Algebra (or Geometric Calculus) but not exterior calculus. Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

17 Beyond the derham Complex... The derivative operator in Clifford algebra is a formal sum of d and its adjoint: := d + δ [0, T ] Λ k+2 d The derham complex appears as diagonals in a full Clifford complex The Bochner complex appears as parametrizations of these diagonals [0, T ] Λ k d δ Λ k+1 Λ k 1 δ d Λ k δ Λ k 2 Finite Element Clifford Algebra will study discretizations of this larger complex. Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18

18 Questions? Slides and pre-prints available at agillette Andrew Gillette - UCSD Finite ( ) Element Clifford Algebra SIAM PD11 - Nov / 18