Schemes. Philipp Keding AT15, San Antonio, TX. Quarklet Frames in Adaptive Numerical. Schemes. Philipp Keding. Philipps-University Marburg

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1 AT15, San Antonio, TX Joint work with Stephan Dahlke and Thorsten Raasch

2 Let H be a Hilbert space with its dual H. We are interested in the solution u H of the operator equation Lu = f, with L : H H a boundedly invertible operator and f H. Example: Poisson equation on Ω = [ 1, 1] 2 \(0, 1] 2 with homogeneous Dirichlet boundary conditions, u = f in Ω, with : H 1 0 (Ω) H 1 (Ω) and f H 1 (Ω).

3 treatment space refinement methods with high convergence rate available (FEM and wavelet methods). There is some hope that we can beat them. Idea: combine adaptivity and space refinement with polynomial enrichment (hp-method). Until now hp-methods exist only in the finite element setting. Quarklets are the first hp approach based on wavelets. In the long run: provable exponential convergence rates.

4 of the operator equation via a frame Ψ = {ψ λ } λ Λ of H leads to a matrix-vector equation Au = f, with A = {Lψ λ (ψ µ )} λ,µ Λ, f = {f (ψ λ )} λ Λ and u = Ψ T u. Note: for a redundant frame the matrix A is only positive semidefinite. Use an adaptive approximation scheme (e.g.: Richardson, steepest descent) to solve this eqution. Essential for high convergence rates: fast convergence in respect of best N-term approximation. certain compression properties of the stiffness matrix A.

5 Construction of To keep it simple: quarklet frames on the real axis. CDF basis. Mother wavelet ψ with ψ(x) = k Z b k ϕ(2x k) for all x R, of order d with d vanishing moments and generator function ϕ = N d ( + d 2 ).

6 Quarks Quarks through multiplying the generator function ϕ with monomes: ( ) x p ϕ p (x) := ϕ(x), for all p 0, x R. m/ (a) p = 0 (b) p = 1 (c) p = 2 Figure: B-Spline Quarks ϕ p of order d = 2

7 Quarklets Quarklets are defined by ψ p (x) = k Z b k ϕ p (2x k) for all p 0, x R (a) p = 0 (b) p = 1 (c) p = 2 Figure: Quarklets ψ p of order d = 2 with d = 2 vanishing moments

8 For j 0 N we analyze systems of dilated and translated ψ p,j,k := 2 j/2 ψ p (2 j k), for all p 0, j j 0, k Z, and quarks ψ p,j0 1,k := ϕ p,j0,k := 2 j 0/2 ϕ p (2 j0 k), for all p 0, k Z.

9 Quarklets have the same amount of vanishing moments as the underlying wavelets. Properly scaled versions of the quarklet systems build frames in L 2 (R) and H 1 (R). The corresponding Laplacian stiffness matrix is compressible (i.e. it can be well aproximated by matrices with finitely many nontrivial entries).

10 Compression property Theorem (Compression of the Laplacian) Let d 3. For J N 0, we define the biinfinite matrix A J by dropping the entries a λ,λ from A when a log 2 (1 + p p ) + b j j > J, (1) with a, b > 0 fulfilling some technical conditions. Then the number of non-zero entries in each row and colum of A J is of order 2 J, and A A J L(l2 (Λ)) 2 J(d 2)/b. (2)

11 results Figure: Compression of the Laplacian A for d = d = 3.

12 Convergence rates 2d (L-shaped domain) Implementation

13 S. Dahlke, P. Keding, T. Raasch: Quarkonial frames with compression properties. Bericht Mathematik Nr des Fachbereichs Mathematik und Informatik, Universität. (Preprint) S. Dahlke, P. Oswald, T. Raasch: A note on quarkonial systems and multilevel partition of unity methods. Math. Nachr. 286 (2013), A. Cohen, I. Daubechies, and J.-C. Feauveau: Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45 (1992),

14 A. Cohen, W. Dahmen, R. DeVore: wavelet methods. II. Beyond the elliptic case. Found. Comput. Math. 2 (2002), R. Stevenson: solution of operator equations using wavelet frames. SIAM J. Numer. Anal. 41 (2003), no. 3, C. Schwab: p- and hp-finite Element Methods. Theory and Applicatios in Solid Fluid Mechanics. Clarendon Press, Oxford, 1998.

15 Thank you very much!