New Model Stability Criteria for Mixed Finite Elements

Transcription

1 New Model Stability Criteria for Mixed Finite Elements Andrew Gillette Department of Mathematics Institute of Computational Engineering and Sciences University of Texas at Austin Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering Mar and 2010 Sciences Universit 1 / 31

2 Outline 1 Introduction and Prior Work 2 Motivation: The DEC-deRham diagram 3 Application: Mixed Finite Element Problems 4 New Stability Criteria for Dual Variables Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering Mar and 2010 Sciences Universit 2 / 31

3 Outline 1 Introduction and Prior Work 2 Motivation: The DEC-deRham diagram 3 Application: Mixed Finite Element Problems 4 New Stability Criteria for Dual Variables Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering Mar and 2010 Sciences Universit 3 / 31

4 Motivation Biological modeling requires stable computational methods to solve PDEs Electromagnetics Electrodiffusion Elasticity These methods must accommodate multiple variables large meshes multi-scale phenomena What does stability mean in such contexts? Computational Visualization Center, I C E S (Department The University of Mathematics of Texas Institute at Austinof Computational Engineering Mar and 2010 Sciences Universit 4 / 31

5 Problem Statement A computational method for solving PDEs should exhibit Model Stability: Computed solutions are found in a subspace of the solution space for the smooth problem Criterion: Solution spaces for variables respect the derham sequence. Discretization Stability: Computed solutions converge as mesh size h decreases and/or polynomial degree of approximation p increases Criterion: The LBB inf-sup condition is satisfied. Numerical Stability: Calculation of the numerical solution has controlled computational complexity. Criterion: Matrices inverted by the linear solver are well-conditioned. Problem Statement Use the theory of Discrete Exterior Calculus to evaluate the stability of existing computational methods for PDEs arising in biology and create novel methods with improved stability. This talk s focus: model stability. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering Mar and 2010 Sciences Universit 5 / 31

6 Selected Prior Work Importance of differential geometry in computational methods for electromagnetics: BOSSAVIT Computational Electromagnetism Academic Press Inc Primer on DEC theory and program of work: DESBRUN, HIRANI, LEOK, MARSDEN Discrete Exterior Calculus arxiv:math/ v2 [math.dg], 2005 Generalization of derham diagram criteria for model stability: ARNOLD, FALK, WINTHER Finite element exterior calculus, homological techniques, and applications Acta Numerica, 15:1-155, Applications of DEC to electromagnetics, Darcy flow, and elasticity problems: HE, TEIXEIRA Geometric finite element discretization of Maxwell equations in primal and dual spaces Physics Letters A, 349(1-4):1 14, 2006 HIRANI, NAKSHATRALA, CHAUDHRY Numerical method for Darcy Flow derived using Discrete Exterior Calculus arxiv: v1 [math.na], 2008 YAVARI On geometric discretization of elasticity Journal of Mathematical Physics, 49(2): , 2008 Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering Mar and 2010 Sciences Universit 6 / 31

7 Outline 1 Introduction and Prior Work 2 Motivation: The DEC-deRham diagram 3 Application: Mixed Finite Element Problems 4 New Stability Criteria for Dual Variables Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering Mar and 2010 Sciences Universit 7 / 31

8 (Smooth) Exterior Calculus Differential k-forms model k-dimensional physical phenomena. u E B q The exterior derivative d generalizes common differential operators. Λ 0 (Ω) d 0 Λ 1 (Ω) grad d 1 Λ 2 (Ω) curl d 2 Λ 3 (Ω) div The Hodge Star transfers information between complementary dimensions. Λ 0 (Ω) Λ 3 (Ω) Λ 1 (Ω) Λ 2 (Ω) Fundamental Theorem of Discrete Exterior Calculus Stable computational methods must recreate the essential properties of smooth exterior calculus on the discrete level. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering Mar and 2010 Sciences Universit 8 / 31

9 Discrete Exterior Calculus Discrete differential k-forms are k-cochains, i.e. linear functions on k-simplices The discrete exterior derivative is D = ( ) T, the transpose of the boundary operator. C 0 D 0 (grad) C 1 D 1 (curl) C 2 D 2 C 3 (div) The discrete Hodge Star M transfers information between complementary dimensions on dual meshes. C 0 M 0 C 1 M 1 C 2 C 1 C 2 M 2 C 0 omputational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering Mar and 2010 Sciences Universit 9 / 31

10 The Importance of Cohomology Λ 0 d 0 grad Λ 1 d 1 curl Λ 2 d 2 div Λ 3 C 0 D 0 C 1 D 1 C 2 D 2 Cohomology classes represent the different types of solutions permitted by the topology of the space. The solution spaces for a discrete method should include representatives from all cohomology classes. Hence model stability requires that the top and bottom sequences have the same cohomology. Example: The torus has two non-zero cohomology classes in dimension 1. C 3 Cohomology at Λ 1 := ker d 1 /im d 0 (if stable) Cohomology at C 1 := ker D 1 /im D 0 Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences10 Universit / 31

11 Mixed finite element methods Mixed finite element methods seek solutions in subspaces of the L 2 derham sequence. H 1 d 0 grad I 0 P 0 H(curl) I 1 P 1 d 1 curl H(div) C 0 D 0 C 1 D 1 C 2 D 2 where I is an interpolation map and P is a projection map. Theorem [Arnold, Falk, Winther] I 2 P 2 d 2 div L 2 I 3 If I k is Whitney interpolation and P k+1 d k = D k P k then the top and bottom sequences have the same cohomology. Proof: The cohomology induced by Whitney interpolation is the simplicial cohomology [Whitney 1957] which is isomorphic to the derham cohomology [derham]. C 3 P 3 Whitney interpolation provides for model stability in simple cases. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences11 Universit / 31

12 The DEC-deRham Diagram for R 3 We combine the Discrete Exterior Calculus maps with the L 2 derham sequence. smooth: H 1 grad I 0 P 0 H(curl) P 1 curl H(div) primal: C 0 D 0 C 1 D 1 C 2 D 2 (M 0 ) 1 dual: C 3 M 0 (D 0 ) T I 1 (M 1 ) 1 C 2 M 1 (D 1 ) T I 2 (M 2 ) 1 C 1 P 2 M 2 div (D 2 ) T L 2 I 3 C 3 (M 3 ) 1 C 0 P 3 M 3 The combined diagram helps elucidate primal and dual formulations of finite element methods. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences12 Universit / 31

13 Outline 1 Introduction and Prior Work 2 Motivation: The DEC-deRham diagram 3 Application: Mixed Finite Element Problems 4 New Stability Criteria for Dual Variables Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences13 Universit / 31

14 Poisson Problem - Primal The smooth Poisson problem on a domain Ω R 3 is ( u = f in Ω u n = 0 on Ω Typical primal discretization: Portion of DEC-deRham diagram: D T 0 M 1 D 0 u = f u D 0 D 0 u M 1 (D 0 ) T M 1 D 0 u M 1 D 0 u (D 0 ) T Ref: W.N. Bell, Dissertation, Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences14 Universit / 31

15 Poisson Problem - Dual ( u = f in Ω u n = 0 on Ω u is a 0-form but it need not be discretized on a primal mesh. From the DEC-deRham diagram, we can derive a dual discretization. Primal discretization: D T 0 M 1 D 0 u = f Dual discretization: D 2 (M 2 ) 1 (D 2 ) T u = f. u D 0 D 0 u M 1 (D 0 ) T M 1 D 0 u M 1 D 0 u (D 0 ) T (M 2 ) 1 (D 2 ) T u (M 2 ) 1 (D 2 ) T u D 2 (D 2 ) T D 2 (M 2 ) 1 (D 2 ) T u The dual discretization may offer improved stability (model, discretization, or numerical). u Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences15 Universit / 31

16 Darcy Flow in R 3 - Primal Flux 8 >< >: f + k p µ = 0 in Ω, div f = φ in Ω, f = ψ on Ω, f C 2 is the volumetric flux through faces of the primal mesh p C 0 is the pressure at vertices of the dual mesh k and µ are constants Mixed (primal + dual) discretization:» (µ/k)m2 D T 2 D 2 0» f p = D 2 f D 2 f» 0 φ. M 2 M 2 f (D 2 ) T p Ref: Hirani, Nakshatrala, Chaudhry, 2008 (D 2 ) T p Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences16 Universit / 31

17 Darcy Flow in R 3 - Dual Flux 8 >< An equally valid discretization is as follows: >: f + k p µ = 0 in Ω, div f = φ in Ω, f = ψ on Ω, f C 2 is the volumetric flux through faces of the dual mesh p C 0 is the pressure at vertices of the primal mesh New mixed discretization:» (µ/k)m 1 1 D 0 (D 0 ) T 0» f p =» 0 φ. p D 0 (M 1 ) 1 f D 0 p (M 1 ) 1 (D 0 ) T f (D f 0 ) T Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences17 Universit / 31

18 Outline 1 Introduction and Prior Work 2 Motivation: The DEC-deRham diagram 3 Application: Mixed Finite Element Problems 4 New Stability Criteria for Dual Variables Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences18 Universit / 31

19 Stability Criteria for Dual Variables The Arnold-Falk-Winther model stability criteria only considers primal discretizations: DEC-based mixed finite element methods require additional criteria for model stability. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences19 Universit / 31

20 Stability Criteria for Dual Variables If we have projection to or interpolation from a dual mesh, we have the maps: smooth: primal: dual: I k Λ k C k P k I n k (M k ) 1 M k C n k P n k Λ n k More concisely, we expect some commuativity of the diagram: smooth Λ k Λ n k smooth primal I k C k P k M k (M k ) 1 I n k C n k P n k dual Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences20 Universit / 31

21 Stability Criteria for Dual Variables smooth Λ k Λ n k smooth primal I k C k P k M k (M k ) 1 I n k C n k P n k dual We identify four subcommutativity conditions: Commutativity at Λ k : M k P k = P n k Commutativity at C k : I k = I n k M k Commutativity at Λ n k : (M k ) 1 P n k = P k Commutativity at C n k : I k (M k ) 1 = I n k To evaluate these conditions, we must now define the various maps involved. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences21 Universit / 31

22 Smooth Hodge Star The smooth Hodge star is defined as the unique map : Λ k Λ n k satisfying the property α β = (α, β) Λ k µ, α, β Λ k denotes the wedge product (, ) Λ k denotes the inner product on k-forms µ is the volume n-form on the domain Example 1: In R 3, let α = β = dx. Then α β = dx dx = dx dydz = µ = (dx, dx) Λ k µ = (α, β) Λ k µ Example 2: In R 3, let α = dx, β = dy. Then α β = dx dy = dx ( dxdz) = 0 = (dx, dy) Λ k µ = (α, β) Λ k µ Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences22 Universit / 31

23 Whitney Interpolation The Whitney k-form η σ k is associated to the k-simplex σ k in the primal mesh. σ 0 := [v i ] η σ 0 := λ i σ 1 := [v i, v j ] η σ 1 := λ i λ j λ j λ i σ 2 := [v i, v j, v k ] η σ 2 := 2 (λ i λ j λ k + λ j λ k λ i + λ k λ i λ j ) 1 on σ σ 3 3 := [v i, v j, v k, v l ] η σ 3 := χ σ 3 = 0 otherwise where λ i denotes the barycentric function for vertex v i. The Whitney interpolant I k of a k-cochain ω, is I k (ω) := X σ k C k ω(σ k )η σ k. Examples of Whitney 1-forms associated to horizontal and vertical edges, respectively Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences23 Universit / 31

24 Commutativity at C k Commutativity at C k : I k = I n k M k Smooth Hodge star: α β = (α, β) Λ k µ, α, β Λ k Whitney interpolant: I k (ω) = X σ k C k ω(σ k )η σ k It suffices to show that for any test function α Λ k α I k = α I n k M k. Check on a basis {ω k i } where ω k i is 1 on σ k i and 0 on all other k-simplices: α I k (ω k i ) = α I n k (M k ω k i ). Use the definitions of I k and to derive the condition: α, η σ ki Λ k µ = α I n k(m k ω k i ). This condition motivates definitions of the dual interpolant I n k and the discrete Hodge star M k that ensure model stability. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences24 Universit / 31

25 Commutativity at C n k Commutativity at C n k : I k (M k ) 1 = I n k Smooth Hodge star: α β = (α, β) Λ k µ, α, β Λ k Whitney interpolant: I k (ω) = X σ k C k ω(σ k )η σ k It suffices to show that for any test function α Λ n k Check on a basis {ω n k i k-simplices: α I n k = α I k (M k ) 1. } where ω n k i is 1 on σ k i and 0 on all other duals of α I n k (ω n k i ) = α I k (M k ) 1 (ω n k i ). Supposing that I n k (ω n k i ) = η σ k, we have i α, η σ ki Λ n k µ = α I k(m k ) 1 (ω n k i ). This complementary condition also motivates definitions of the dual interpolant I n k and the discrete Hodge star M k. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences25 Universit / 31

26 Criteria Applied to Darcy Flow - Dual Flux p D 0 (M 1 ) 1 f D 0 p (M 1 ) 1 (D 0 ) T f (D f 0 ) T We check for commutativity of the pressure data, i.e. at C 0 with n = 3, k = 0: α, η σ 0i H 1 µ = α I 3(M 0 ω 0 i ) α H 1 We use the Hodge star proposed by the authors of the paper (M 0 ) ii := σk i σ k i We use any dual interpolant I 3 mimicking Whitney forms, i.e. I 3 (ω) := X ω( σ 0 )χ σ 0 σ 0 C 3 Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences26 Universit / 31

27 Criteria Applied to Darcy Flow - Dual Flux The left side: The right side: α, η σ 0i µ = (α, λ i) H 1 H 1 µ Z «= αλ i + α λ i µ α I 3 (M 0 ω 0 i ) = α X K σ 0 C 3 (M Diag 0 ω i )( σ 0 )χ σ 0µ = α σ 0 i χ σ 0 i µ = α σ 0 i χ σ 0 i µ The condition: Z K αλ i + α λ i «µ = α σ 0 i χ σ 0 i µ α H 1 Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences27 Universit / 31

28 Criteria Applied to Darcy Flow - Primal Flux D 2 f D 2 f M 2 M 2 f (D 2 ) T p (D 2 ) T Similarly, we can check for commutativity at C 0 for the primal flux version. Z «α(x)λ i (x)dx µ = α σi 3 χ σ 3µ α L 2 i K p Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences28 Universit / 31

29 Criteria Applied to Darcy Flow - Conclusions Dual flux condition: Z Primal flux condition: Z K «αλ i + α λ i µ = α σi 0 χ σ 0µ α H 1 i K «α(x)λ i (x)dx µ = α σ 3 i χ σ 3 i µ α L 2 In both instances, an arbitrary test function α must be approximately constant on a neighborhood of vertex i and this constant is a multiple of a measure of the region and an integral involving α. This is certainly false in general, as L 2 or H 1 functions need not be locally constant. Hence, the diagonal Hodge star espoused by the authors does not provide a stable method in the general setting, in either of the possible mixed finite element methods. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences29 Universit / 31

30 Additional Research Directions Evaluate alternate definitions of M k for our model stability commutativity conditions. Define dual interpolants I k mimicking Whitney primal interpolants. Define M 1 k using dual interpolants to avoid inverting sparse matrices. Compare discretization and numerical stability for these definitions to existing methods. Derive model stability criteria for the elasticity complex which involves vector-valued and tensor-valued forms. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences30 Universit / 31

31 Questions? Slides available at Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational EngineeringMar and2010 Sciences31 Universit / 31