A Generalization for Stable Mixed Finite Elements

Transcription

1 A Generalization for Stable Mixed Finite Elements Andrew Gillette joint work with Chandrajit Bajaj Department of Mathematics Institute of Computational Engineering and Sciences University of Texas at Austin, USA Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM and SepSciences 00 University / 0

2 Motivation Biological modeling requires robust computational methods to solve integral and differential equations over spatially realistic domains. Electrostatics Electromagnetics/ Electrodiffusion Elasticity These methods must accommodate complicated domain geometry and topology multiple variables and operators Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austin of Computational Engineering SPM and Sep Sciences 00 University / 0

3 Notions of Robustness A robust computational method for solving PDEs should exhibit Model Conformity: Computed solutions are found in a subspace of the solution space for the continuous problem Criterion: Discrete solution spaces replicate the the derham sequence. Discretization Stability: The true error between the discrete and continuous solutions is bounded by a multiple of the best approximation error Criterion: The discrete inf-sup condition is satisfied. Bounded Roundoff Error: Accumulated numerical errors due to machine precision do not compromise the computed solution Criterion: Matrices inverted by the linear solver are well-conditioned. Problem Statement Use the theory of Discrete Exterior Calculus to evaluate the robustness of existing computational methods for PDEs arising in biology and create novel methods with improved robustness. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM and SepSciences 00 University / 0

4 Outline Basics of Discrete Exterior Calculus Alternative Discretization Pathways Stability Criteria from Discrete Hodge Stars Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM and SepSciences 00 University 4 / 0

5 Outline Basics of Discrete Exterior Calculus Alternative Discretization Pathways Stability Criteria from Discrete Hodge Stars Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM and SepSciences 00 University / 0

6 (Continuous) Exterior Calculus Differential k-forms model k-dimensional physical phenomena. u E B q The exterior derivative d generalizes common differential operators. H d 0 H(curl) grad d H(div) curl d L div The Hodge Star transfers information between complementary dimensions of primal and dual spaces. H L H(curl) H(div) Fundamental Theorem of Discrete Exterior Calculus Conforming computational methods must recreate the essential properties of (continuous) exterior calculus on the discrete level. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM and SepSciences 00 University 6 / 0

7 Discrete Exterior Calculus Discrete differential k-forms are k-cochains, i.e. linear functions on k-simplices cochain cochain cochain The discrete exterior derivative D is the transpose of the boundary operator D 0 4 matrix with entries 0, ± This creates a discrete analogue of the derham sequence. C 0 D 0 (grad) C D (curl) C D C (div) Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM and SepSciences 00 University 7 / 0

8 Discrete Exterior Calculus The discrete Hodge Star M transfers information between complementary dimensions on dual meshes. In this example, we use the identity matrix for M M matrix The discrete exterior derivative on the dual mesh is D T D T + 4 matrix with entries 0, ± This creates a dual discrete analogue of the derham sequence C D T (div) C D T (curl) C D T 0 C 0 (grad) Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM and SepSciences 00 University 8 / 0

9 The DEC-deRham Diagram for R We combine the Discrete Exterior Calculus maps with the L derham sequence. derham: H grad I 0 P 0 H(curl) P curl H(div) primal: C 0 D 0 C D C D (M 0 ) dual: C M 0 (D 0 ) T I (M ) C M (D ) T I (M ) C P M div (D ) T L I C (M ) C 0 P M The combined diagram elucidates alternative discretization pathways for finite element methods. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM and SepSciences 00 University 9 / 0

10 Outline Basics of Discrete Exterior Calculus Alternative Discretization Pathways Stability Criteria from Discrete Hodge Stars Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences0 University / 0

11 Linear Poisson-Boltzmann Equation div `ɛ( x) φ( x) = ρ c( x) + κ( x)φ( x)in R φ( x) = electrostatic potential j ɛi, x Ω ɛ( x) = dielectric coefficient = ɛ E, x R Ω ρ c( x) = charge density from atomic charges κ( x) = modified Debye-Huckel parameter Exterior calculus formulation: d ɛdφ = f, φ H, f L Primal discretization: D T 0 M ɛd 0 φ = f Dual discretization: D ɛ(m ) (D ) T φ = f φ D 0 ɛd 0 φ M (D 0 ) T M ɛd 0 φ M ɛd 0 φ (D 0 ) T ɛ(m ) (D ) T φ (M ) (D ) T φ D D ɛ(m ) (D ) T φ φ (D ) T Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences University / 0

12 DEC for LPBE The two discretizations inside the DEC-deRham diagram: H I 0 φ P 0 grad D 0 (D 0 ) T H(curl) I P ɛd 0 φ M (D 0 ) T M ɛd 0 φ M ɛd 0 φ curl H(div) I P ɛ(m ) (D ) T φ (M ) (D ) T φ div D (D ) T L I P D ɛ(m ) (D ) T φ φ The definition of the discrete Hodge star matrix M k and its inverse are essential in ensuring the robustness of a primal or dual discretization. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences University / 0

13 Further Examples Maxwell s Curl Equations µ E = ω ɛ E ɛ h = ω H Darcy Flow 8 >< >: D primal: E C (M ) dual: C M f + k p µ = 0 in Ω, div f = φ in Ω, f ˆn = ψ on Ω, (D ) T (M ) H M primal: p D 0 D 0 p (M Dual ) f (M Dual ) dual: (D 0 ) T f f (D 0 ) T D f D f M Diag M Diag f (D ) T p (D ) T p Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences University / 0

14 Outline Basics of Discrete Exterior Calculus Alternative Discretization Pathways Stability Criteria from Discrete Hodge Stars Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences4 University / 0

15 Discrete Hodge Star Criteria A discrete Hodge star transfers information between primal and dual meshes: primal mesh simplex σ k dual mesh cell σ k DIAGONAL [Desbrun et al.] (M Diag k ) ij := σk i δ σj k ij WHITNEY [Dodziuk],[Bell] (M Whit k ) ij := η σ, η ki σ (η σ kj C k k = Whitney k-form for σ k ) A robust definition of a discrete Hodge star matrix M k and its inverse should provide for: Commutativity of discrete dual operators Local structure of M k and M k Well-conditioned matrices Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences University / 0

16 Commutativity of Discrete Dual Operators Given projection to (P) or interpolation from (I) a dual mesh, we have the maps: derham: primal: dual: I k Λ k C k P k I n k (M k ) M k C n k P n k Λ n k Thus, we expect some commutativity of the diagram: derham Λ k Λ n k derham primal I k C k P k M k (M k ) 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 Engineering SPM Sep and00 Sciences6 University / 0

17 Commutativity of Discrete Dual Operators derham Λ k Λ n k derham primal I k C k P k I n k M k (M k ) C n k P n k dual Strong commutativity at C k : Z I k = IZ n k M k Weak commutativity at C k : α I k = α I n k M k, T T α Λ k Example: Discrete Hodge star definitions can be evaluated by this criteria: Z Using M Diag 0 : T (α, λ i ) H = σi 0 αµ, α H σ i 0 Using M Whit 0 : T (α, λ i ) H = X Z (λ i, λ j ) H αµ, α H vertex j σ 0 i Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences7 University / 0

18 Local Structure of M k and M k Existing inverse discrete Hodge stars are either too full or too empty for use in discretizations on dual meshes We present a novel dual discrete Hodge star for this purpose using generalized barycentric coordinate functions (details in the paper) type definition M k M k DIAGONAL (M Diag k ) ij = σk i δ σj k ij diagonal diagonal WHITNEY DUAL (M Whit k ) ij = η σ, η ki σ sparse (full) kj C k ((M Dual k ) ) ij = η σ ki, η σ kj C k (full) sparse Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences8 University / 0

19 Well-Conditioned Matrices The condition number of a discrete Hodge star matrix depends on the size of both primal and dual mesh elements. Primal simplices σ k satisfy geometric quality measures. Dual cells σ k satisfy geometric quality measures. The value of σ k / σ k is bounded above and below. 4 The primal and dual meshes do not have large gradation of elements. We identify which geometric criteria are required to provide bounded condition numbers on the various discrete Hodge star matrices. DIAGONAL WHITNEY DUAL i ii iii iv M Diag k M Whit k (M Dual k ) Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences9 University / 0

20 Questions? Slides available at This talk was presented at the 00 Symposium of Solid and Physical Modeling in Haifa, Israel. This research was supported in part by NIH contracts R0-EB00487, R0-GM0748, and a grant from the UT-Portugal CoLab project. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering SPM Sep and00 Sciences0 University / 0