Web-Spline Approximation of Elliptic Boundary Value Problems Ulrich - PowerPoint PPT Presentation

1 Web-Spline Approximation of Elliptic Boundary Value Problems Ulrich Reif Darmstadt University of Technology ❆ Klaus H¨ ollig Joachim Wipper University of Stuttgart Presented by U. Reif at the the Fifth International Conference on Mathematical Methods for Curves and Surfaces, Oslo, July 4, 2000.

2 Overview ❏ Model problem ❏ Standard FE-techniques ❏ Uniform b-splines ❏ Weighted extended b-splines • Stability • Approximation order ❏ Examples ❏ Multigrid ❏ Extensions and further development ❏ Conclusion

3 Model problem R m On a bounded domain Ω ⊂ I we consider Poisson’s equation − ∆ u = f in Ω with Dirichlet boundary conditions u = 0 on ∂ Ω . Ω

4 Weak formulation: � � ∀ ψ ∈ H 1 ∇ u ∇ ψ = fψ, 0 . Ω Ω B = span { B i , i ∈ I } An approximation in a finite dimensional subspace I � a i B i ≈ u ∈ H 1 B ∋ u h = I 0 i ∈ I is obtained by solving the Galerkin system � � � ∇ B k ∇ B i a i = fB k , k ∈ I Ω Ω i ∈ I � k ∈ I g k,i a i = f k , i ∈ I GA = F

5 Objectives: ❏ fast convergence u h → u as h → 0 convergence u h → u as h → 0 ❏ respect boundary conditions ❏ cond G h ∼ h − 2 ❏ low dimensional subspace ❏ efficiency, i.e. number of iterations ∼ 1 /h or even ∼ 1 ❏ practicability

6 Standard FE-techniques mesh-based: ❏ hat functions ❏ macro elements (Clough-Tocher, Agyris, Schumaker) meshless: ❏ radial basis functions ❏ wavelets ❏ hp elements

7 Hat functions: ❏ Based on triangulation (or quadrangulation) of Ω . ❏ 2d-meshing expensive. Figures by Dietrich Nowottny

8 Hat functions: ❏ Based on triangulation of Ω . ❏ 2d-meshing expensive. ❏ 3d-meshing very expensive. Figures by Alexander Fuchs

9 Hat functions: ❏ Based on triangulation of Ω . ❏ 2d-meshing expensive. ❏ 3d-meshing very expensive. ❏ Slow convergence, � u − u h � 0 ∼ h 2 . ❏ High dimensional subspaces, B ∼ � u − u h � − m/ 2 dim I . 0 ❏ cond G h ∼ h − 2 , iff triangulation is uniform. ❏ Huge amount of code implemented and optimized.

10 Meshless methods: unstructured structured Main difficulties: • Obey boundary conditions. • Obey boundary conditions. • Control condition number.

11 Babu˘ ska proposes: ❏ Lagrange multiplier method • saddle point problem • indefinite system • LBB condition ❏ Penalty method • minimize energy + penalty on boundary deviation • balance of terms very delicate ”Both methods have their adherents, . . . , none, however, has gained universal popularity”(Bochev & Gunzberger ’98).

12 Uniform b-splines Z m is The tensor product b-spline basis of order n with knots h Z Z m } , supp b k = h ( k + [0 , n ] m ) . { b k : k ∈ Z Potential benefit: ❏ No mesh generation required. ❏ Fast convergence, � u − u h � 0 ∼ h n . ❏ Low (lowest) dimensional subspace B ∼ � u − u h � − m/n dim I . 0

13 Problems: ❏ Boundary conditions: • If a spline is zero on the boundary of Ω , then it vanishes on all intersecting grid cells (in general). This implies a complete loss of approximation power. • Apply Babu˘ ska methods?

14 Problems (contd.): ❏ Condition number: • b-splines with small support in Ω may lead to excessively large condition numbers. • Leaving out outer b-splines reduces approximation power. • Just ignore it (brute force)? 40 10 30 10 condition number 20 10 10 10 0.02 0.04 0.06 0.08 0.1 grid width h

15 Weighted extended b-splines (web-splines) Z m : supp b k ∩ Ω � = ∅} : Partition relevant indices K := { k ∈ Z The inner b-splines with indices The outer b-splines with indices I ⊂ K J = K \ I have at least one grid cell in have no grid cell in their support their support contained in Ω . contained in Ω .

16 Extension: In order to stabilize the basis, the outer b-splines are no longer considered to be independend. Instead, they are coupled with inner b-splines, � i ∈ I. B i = b i + e i,j b j , j ∈ J ❏ B i is an extended b-spline, i.e. supp B i ⊃ supp b i . ❏ Local extension yields uniformly bounded support, e i,j = 0 for � i − j � � 1 ⇒ | supp B i | � h. Moreover, most b-splines remain unchanged. ❏ Choose coefficients e i,j in such a way that all polynomials of order n remain in the span of the extended B-Splines B i using Marsden’s identity, � p ( k ) b k ∈ I p ∈ I P n (Ω) iff P n ( K ) . k ∈ K

17 For any outer index j ∈ J let • I ( j ) ⊂ I be a closest inner array of dimension n m , • J ( i ) = { j ∈ J : i ∈ I ( j ) } be the dual index set of I ( j ) . • L i , i ∈ I ( j ) , be the Lagrange po- lynomials associated with I ( j ) .

18 For any outer index j ∈ J let Choosing the coefficients � • I ( j ) ⊂ I be a closest inner array for i ∈ I ( j ) L i ( j ) e i,j = of dimension n m , 0 else • J ( i ) = { j ∈ J : i ∈ I ( j ) } be the yields the wanted representation dual index set of I ( j ) . � � • L i , i ∈ I ( j ) , be the Lagrange po- p ( i ) B i = p ( k ) b k . lynomials associated with I ( j ) . i ∈ I k ∈ K 3 −3 1 0 0 0 0 0 0 0 1 3 0 0 3

19 Weighting: The incorporation of zero boundary conditions is amazingly simple. R + Let w : Ω → I 0 be a smooth function equivalent to the boundary distance, i.e. w ( x ) dist( x, ∂ Ω) dist( x, ∂ Ω) � 1 , � 1 , w ( x ) and in particular w = 0 exactly on ∂ Ω . Multiplying the extended b-splines B i by the weight function w yields a basis which satisfies the boundary condition.

20 Definition: The web-splines B i are defined by � � � w B i = b i + e i,j b j , i ∈ I, w ( x i ) j ∈ J ( i ) where x ( i ) is the center of a grid cell in supp b i ∩ Ω . The web-splines span the web-space B := span { B i : i ∈ I } . I

21 Stability For λ k , k ∈ I, a family of dual functionals for b i supported on Ω let Λ k = w ( x k ) λ k . w Theorem 1: For i, k ∈ I , the dual functionals Λ k and the web- splines B i are uniformly bounded in L 2 with respect to the grid width h , and biorthogonal, � � B i � 0 � 1 , � Λ k � 0 � 1 , B i Λ k = δ i,k . Ω Theorem 2: The web-basis is stable with respect to the L 2 -norm, � � � � � 0 ∼ � A � . a i B i � � i ∈ I

22 Theorem 3: The web-basis satisfies � � � � � r � h − r � A � . a i B i � � i ∈ I Theorem 4: The spectrum of the Galerkin matrix G h is bounded by 1 � ̺ ( G h ) � h − 2 . Theorem 5: The condition number of the Galerkin matrix is bounded by cond G h � h − 2 .

23 Approximation order Theorem 6: Let u ∈ H 1 0 be a smooth function. Then � �� � � u − v h � r � h n − r , v h = P u := u Λ i B i . i ∈ I Theorem 7: Let u be a smooth solution of the model problem and u h ∈ I B a finite element approximation obtained by solving the Galerkin system. Then � u − u h � r � h n − r .

24 Multigrid The performance of cg-solvers ( ∼ h − 1 iterations) can be improved by multigrid methods. These require ❏ a smoothing operator S , e.g. Richardson’s method S : A → A + λ − 1 max ( F − GA ) . B 2 h → I B h , ❏ a grid transfer operator P : I P : A 2 h → A h = PA 2 h with matrix entries � � � p ℓ,i = w ( x h ℓ ) e 2 h c ℓ − 2 i + i,j c ℓ − 2 j . w ( x 2 h i ) j ∈ J 2 h ( i )

25 Multigrid Algorithm U → W = M ( U, F, h ) : V = S α U % α smoothing iterations � F = P t ( F − GV ) % residual on coarse grid if 2 h = h max % G − 1 � W = � � F % direct solution on coarsest grid else % W = M β (0 , � � F, 2 h ) % β multigrid steps end % W = V + P � W % update on fine grid Theorem 8: For β = 2 and α sufficiently large (W-cycle), the multigrid algorithm converges after O (1) iterations. Thus, the complexity for solving the FE-problem reduces to O (dim I B) .

26 Extensions and further development ❏ The method potentially applies to many FE problems. ❏ Hierarchical b-splines can be used for local and adaptive grid refinement. ❏ The weight function is still subject to optimization. ❏ Extend the method to non-smooth problems • by local refinement, • by assymptotic expansion. ❏ Implementation (3d, multigrid) in progress.

27 Conclusion The web-spline method is a promising new FE technique providing the following features: ❏ Wide range of applicability. ❏ No mesh generation required. ❏ High accuracy approximation with relatively few coefficients. ❏ O (1) -convergence with multigrid. ❏ Based on industrial standard (b-splines). ❏ Easy to implement (3d integration subtle).