Finite Element Clifford Algebra: A New Toolkit For .
Finite Element Clifford Algebra: A New Toolkit forEvolution ProblemsAndrew Gillettejoint work with Michael HolstDepartment of MathematicsUniversity of California, San Diegohttp://ccom.ucsd.edu/ agillette/Andrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 20111 / 18
MotivationPoisson’s equation: Given f find u(x) such that(0 u f in Ω Rnu 0aon ΩHeat equation: Given f and g, find u(x, t) such that ut u f in Ω Rn , u 0on Ω, u t 0 gin Ωfor t 0,for t 0,Finite element exterior calculus (FEEC) provides:abstract framework for analyzing numerical approximation of elliptic PDEsclassification of stable finite element methods with optimal convergence ratesHow can the FEEC framework be expanded to classify stable finite element methodsfor evolutionary PDEs?Andrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 20112 / 18
Outline of approachTwo possible methods for extending Finite Element Exterior Calculus:Semi-discrete: Finite element method in space, ODE in timedomain: tΩΩ [0, T ] Rn Rsolution basis:φh t t0 : Ω R, for each t0 [0, T ]error analysis:FEEC Bochner space theoryFully discrete: Finite element method in space and timeΩdomain:Ω [0, T ] Rn Rsolution basis:φh : Ω [0, T ] Rerror analysis:Finite Element Clifford AlgebraThis talk: Initial results on semi-discrete approach a preview of FECAAndrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 20113 / 18
Table of Contents1Motivation2Background: FEEC, Bochner Spaces, Semi-Discrete methods3Results: New error estimates in Bochner norms4Preview: Why FECA is neededAndrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 20114 / 18
Outline1Motivation2Background: FEEC, Bochner Spaces, Semi-Discrete methods3Results: New error estimates in Bochner norms4Preview: Why FECA is neededAndrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 20115 / 18
Finite Element Exterior Calculus in contextConsider a mixed method for Poisson’s problem on a domain Ω Rn :continuous u f 0,u H2mixed weak(div σ, φ) (f , φ) 0, φ L2 Λn(σ, ω) (u, div ω) 0, ω H(div) Λn 1(div σh , φh ) (f , φh ) 0, φh Λnh L2(σh , ωh ) (uh , div ωh ) 0, ωh Λn 1h H(div)mixed FEMMajor Conclusions from FEECThe finite elements spaces Λn 1and Λn 1should be chosen from two classes ofhhpiecewise polynomial spaces, denoted Pr Λkh and Pr ΛkhIf this choice is made in a compatible manner implied by the exterior calculusstructure, then optimal a priori error estimates are guaranteedAndrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 20116 / 18
Finite Element Exterior Calculus in contextTheorem [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 sholds for 0 s smax . Choose finite element spaces n n 1 Pr 1 Λ (T ) Pr 1 Λ (T ) nn 1oror, Λh Λh n 1Pr Λn (T )Pr 1 Λ (T )Then for 0 s smax , the following error estimates hold(ch f L2if Λnh P1 Λn (T ), u uh L2 ch2 s f H s otherwise,( σh σ L2 chs 1 f H s div (σh σ) L2 chs f H s ,Andrew Gillette - UCSDififif s r 1Λn 1 Pr 1 Λn 1 ,hs r 1,Λn 1 Pr 1 Λn 1 ,hs r,s r 1.Finite( ) Element Clifford AlgebraSIAM PD11 - Nov 20117 / 18
Semi-discrete Mixed FormulationConsider a mixed method for the heat equation on Ω Rn for t I : [0, T ].ut u f,u t 0 g.(ut , φ) (div σ, φ) (f , φ),continuousmixed weakmixed FEM φ Λn , ω Λn 1t I,t I,(σ, ω) (u, div ω) 0,u t 0 g.(uh,t , φh ) (div σh , φh ) (f , φh ), φh Λnh ,t I,(σh , ωh ) (uh , div ωh ) 0, ωh Λn 1,ht I,uh t 0 gh .AUt BΣ F 0linear systemTB U DΣAndrew Gillette - UCSDFinite( ) Element Clifford Algebra, AUt BD 1 B T U FSIAM PD11 - Nov 20118 / 18
Semi-discrete Error BoundsTheorem [Thomée; Galerkin FEM for Parabolic Problems, 1997]Fix n 2 and set Λ2h : discontinuous linear, Λ1h : Raviart-Thomas elements.Let gh be the solution to the ellipticproblem with f g. Then for t 0:Z t2 uh (t) u(t) L2 ch u(t) H 2 ut H 2 ds ,0 Z 1/2 ! σh (t) σ(t) L2 ch2t u(t) H 3 0 ut 2H 2 ds.Homogeneous case (f 0), gh as above, t 0: uh (t) u(t) L2 ch2 g H 2 ,if g Ḣ 2 , σh (t) σ(t) L2 ch3 g H 3 ,if g Ḣ 3 .Homogeneous case (f 0), gh : orthogonal projection of g on to Λ2h , t 0: uh (t) u(t) L2 ch2 t 1 g L2 σh (t) σ(t) L2 ch2 t 3/2 g L2Note: These bounds are ‘space-only’ and restricted to the case n 2.Andrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 20119 / 18
Bochner spaces and normsOur new error bounds will employ the theory of Bochner spacesDefinitionLet X be a Banach space and I (0, T ). DefineC(I, X ) : {u : I X u bounded and continuous}Equip this space with the norm u C(I,X ) : sup u(t) X .t IPThe Bochner space L (I, X ) is defined to be the completion of C(I, X ) with respect tothe norm: Z 1/p u Lp (I,X ) : I u(t) pX dt.We combine notations to get Bochner differential form spaces:L2 Xk : L2 (I, L2 Λk (Ω))These are parametrized differential form spaces.Andrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 201110 / 18
Outline1Motivation2Background: FEEC, Bochner Spaces, Semi-Discrete methods3Results: New error estimates in Bochner norms4Preview: Why FECA is neededAndrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 201111 / 18
Bochner-FEEC Parabolic Error EstimatesWe 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 sholds for 0 s smax and t I. Choose finite element spaces n n 1 Pr 1 Λ (T ) Pr 1 Λ (T ) n 1nororΛh , Λh n 1 Pr Λn (T )Pr 1 Λ (T )Then for 0 s smax and gh the solution to the elliptic problem we have if Λnh P1 Λn (T ) ch f L2 (I,L2 ) T ft L1 (I,L2 ) uh u L2 Xn ch2 s f T ft L1 (I,H s )otherwise, if s r 1L2 (I,H s ) and. . .Andrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 201112 / 18
Bochner-FEEC Paraoblic Error EstimatesTheorem [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 sholds for 0 s smax and t I. Choose finite element spaces n n 1 Pr 1 Λ (T ) Pr 1 Λ (T ) n 1noror, Λh Λh n 1 Pr Λn (T )Pr 1 Λ (T )Then for 0 s smax and gh the solution to the elliptic problem we have: n 1 n 1 Λh P1 Λ (T ), s 1 orand Λnh P1 Λn (T ), thenIf n 1 n 1Λh P1 Λ (T ), s 0 σh σ L2 Xn 1 c h1 s f L2 (I,H s ) h T ft L2 (I,L2 )For any other choice of spaces, if s r 1, σh σ L2 Xn 1 c h1 s f L2 (I,H s ) h2 s T ft L2 (I,L2 )Andrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 201113 / 18
Proof and SignificanceKey idea of the proof: u(t) uh (t) L2{z} error between weak andsemi-discrete u(t) ũh (t) L2 {z}error between weak andtime-ignorant elliptic ũh (t) uh (t) L2 {z}error between time-ignorantelliptic and semi-discreteSignificance of the error estimatesThese results give a priori estimates of convergence rates for the semi-discreteGalerkin FEM for the heat equation.By using the FEEC framework, we have classified choices of semi-discrete finiteelement 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 findstronger error estimates akin to Thomée’s.G, H OLST, Finite Element Exterior Calculus for Evolution Problems, in preparation.Andrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 201114 / 18
Outline1Motivation2Background: FEEC, Bochner Spaces, Semi-Discrete methods3Results: New error estimates in Bochner norms4Preview: Why FECA is neededAndrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 201115 / 18
The Bochner ComplexFEEC theory studies discretizations of the L2 deRham complex:0/ HΛ0dΩ(grad)/ HΛ1dΩ/ ···dΩ(div)/ HΛndΩ/0We can define a parametrized exterior derivative operator on Bochner spaces:d : HXk HXk 1where (dµ)(t) : dΩ (µ(t)).This gives rise to a Bochner domain complex:0/ HX0d/ HX1d/ ···d/ HXnd/0For a ‘fully discrete’ method, we need an exteriorderivative operator on spacetime elements whichcan distinguish spacelike and timelike dimensions.ΩAndrew Gillette - UCSDSuch an operator needs the Lorentzian signature ofbasis elements - a tool available in Clifford Algebra(or Geometric Calculus) but not exterior calculus.Finite( ) Element Clifford AlgebraSIAM PD11 - Nov 201116 / 18
Beyond the deRham Complex. . .The ‘derivative’ operator in Clifford algebra is a formal sum of d and its adjoint: : d δ6Λ[0, T ]k 2d The deRham complexappears as diagonals ina full ‘Clifford complex’[0, T ]The Bochner complexappears asparametrizations ofthese diagonalsΛk6Λk 1 d δ(6 Λk dδ(Λk 1 δ(Λk 2Finite Element Clifford Algebra will study discretizations of this larger complex.Andrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 201117 / 18
Questions?Slides and pre-prints available at http://ccom.ucsd.edu/ agilletteAndrew Gillette - UCSDFinite( ) Element Clifford AlgebraSIAM PD11 - Nov 201118 / 18
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 2011 16 / 18. Beyond the deRham Complex::: The ‘derivative’ operator rin Clifford algebra is a formal sum of d and its adjoint:
Finite element analysis DNV GL AS 1.7 Finite element types All calculation methods described in this class guideline are based on linear finite element analysis of three dimensional structural models. The general types of finite elements to be used in the finite element analysis are given in Table 2. Table 2 Types of finite element Type of .
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2.2. Linear property of 2-vectors. The Clifford algebra is not a graded multi vector algebra, however it is a graded vector space [4]: it can be decomposed as sum of linear subspaces of homogeneous grade. The twists, or 2-vectors, form a vector subspace within the Clifford algebra over the scalars (0-vectors) and also over the dual scalars,
1 Overview of Finite Element Method 3 1.1 Basic Concept 3 1.2 Historical Background 3 1.3 General Applicability of the Method 7 1.4 Engineering Applications of the Finite Element Method 10 1.5 General Description of the Finite Element Method 10 1.6 Comparison of Finite Element Method with Other Methods of Analysis
Finite Element Method Partial Differential Equations arise in the mathematical modelling of many engineering problems Analytical solution or exact solution is very complicated Alternative: Numerical Solution – Finite element method, finite difference method, finite volume method, boundary element method, discrete element method, etc. 9
3.2 Finite Element Equations 23 3.3 Stiffness Matrix of a Triangular Element 26 3.4 Assembly of the Global Equation System 27 3.5 Example of the Global Matrix Assembly 29 Problems 30 4 Finite Element Program 33 4.1 Object-oriented Approach to Finite Element Programming 33 4.2 Requirements for the Finite Element Application 34 4.2.1 Overall .
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