Besov regularity for operator equations on patchwise smooth manifolds

Transcription

1 on patchwise smooth manifolds Markus Weimar Philipps-University Marburg Joint work with Stephan Dahlke (PU Marburg) Mecklenburger Workshop Approximationsmethoden und schnelle Algorithmen Hasenwinkel, March 204 Research supported by Deutsche Forschungsgemeinschaft DFG (DA 360/9-)

2 Outline Patchwise smooth manifolds Boundary integral : double layer New Besov-type spaces for the double layer equation

3 Patchwise smooth manifolds Here: Lipschitz surfaces Ω which are boundaries of bounded, simply connected, closed polyhedra Ω R 3 with finitely-many flat, quadrilateral sides and straight edges patchwise decomposition: I Ω = F i i= local description as boundary C n of tangent cones C n subordinate to vertices ν n of Ω

4 Boundary integral : double layer Indirect methods for PDE s in Ω (or Ω c ) like U = 0, U = g on Ω, naturally lead to boundary integral such as ( ) S DL (v) := 2 Id K (v) = g on Ω. () Therein K denotes the harmonic double layer v K(v) := v(x) 4π Ω ds(x). η(x) x 2

5 We like to solve the double layer eq. () numerically using (adaptive) wavelet Galerkin boundary element methods. Question Does adaptivity pay off here?

6 We like to solve the double layer eq. () numerically using (adaptive) wavelet Galerkin boundary element methods. Question Does adaptivity pay off here? A rule of thumb for (elliptic) PDE s: On smooth domains there is no need for adaptivity On (general) Lipschitz domains adaptive schemes outperform uniform schemes because in the second case the Besov smoothness of the solution is significantly higher than its Sobolev smoothness. Question How about boundary integral (or even more general operator )?

7 Weighted on Ω For k N and 0 ϱ k define X k ϱ ( Ω) := C patchwise ( Ω) X k ϱ ( Ω) with norm u Xϱ k ( Ω) := N ϕ n u Xϱ k ( C n ) such that Therein δn k ϱ k f n= 2 f Xϱ k ( C n ). (ϕ n ) N n=... special resolution of unity on Ω subordinate to vertices ν n δ n... distance to interfaces k... vector of kth order derivatives (cf. Elschner, Maz ya et al., Babuska, Kondratiev,... )

8 Wavelet bases and (unweighted) on Ω Using parametric liftings κ i : [0, ] 2 F i define an (equivalent) inner product on L 2 ( Ω) by I u, v := u κ i, v κ i L2 ([0,] 2 ). i= Moreover consider, -biorthogonal wavelet Riesz bases Ψ = (Ψ Ω, Ψ Ω ) which characterize L 2 ( Ω), i.e. u L 2 : u = P j (u) + Ω u, ψ j,ξ ψj,ξ Ω j j with u 2 P j (u) 2 + ξ Ω j j j ξ Ω j u, ψ Ω j,ξ /2 2.

9 In addition, Ψ needs to satisfy a couple of (quite technical) conditions, e.g. normalization and support conditions interior vanishing moments of order d N: Ω P, ψ j,ξ κ i L = 0 P 2 ([0,] 2 ) Π d ([0, ] 2 ) and all (j, ξ) such that supp ψ Ω j,ξ F i for some i H s -norm equivalences for /2 < s < min{3/2, s Ω }: u H s P j (u) 2 + j j 2 j s 2 ξ Ω j u, ψ Ω j,ξ /2 2 Typical example which satisfies all requirements: composite wavelet basis (cf. Dahmen/Schneider)

10 Classical B α q (L p (R d )) essentially generalize Sobolev (Hilbert) spaces H s.... depend on (at least) 3 parameters: α, p, q.... are defined in various ways (e.g. using harmonic analysis, moduli of smoothness, interpolation,... ).... are characterized by decay properties of expansion coefficients w.r.t. various building blocks (atoms, quarks, wavelets,... ).

11 Classical B α q (L p (R d )) essentially generalize Sobolev (Hilbert) spaces H s.... depend on (at least) 3 parameters: α, p, q.... are defined in various ways (e.g. using harmonic analysis, moduli of smoothness, interpolation,... ).... are characterized by decay properties of expansion coefficients w.r.t. various building blocks (atoms, quarks, wavelets,... ). Corresponding spaces can also be defined for domains and (nonsmooth) manifolds (as trace spaces or via pullbacks), BUT currently there seems to exist no approach, suitable for numerical applications, to define higher order (Besov) smoothness for functions on patchwise smooth manifolds!

12 New Besov-type spaces on Ω (Dahlke, W. 203) A tuple of real parameters (α, p, q) is called admissible if 2 p α 2 + and 0 < q. 2 By BΨ,q α (L p( Ω)) we denote the set of all u L 2 ( Ω) s.t. the (quasi-) norm u BΨ,q α (L p( Ω)) defined by P j (u) L p ( Ω) ( [ + 2 j α+2 ]) p 2 q j j ξ Ω j u, ψ Ω j,ξ q/p /q p (with the usual modifications for q = ) is finite.

13 always quasi-banach spaces (Banach min{p, q} / Hilbert p = q = 2) simplified (quasi-) norms for so called adaptivity scale p = q = τ := (α/2 + /2) P j (u) L τ ( Ω) + j j and for Hilbert scale p = q = 2 P j (u) 2 + j j 2 j α 2 ξ Ω j ξ Ω j u, u, ψ Ω j,ξ ψ Ω j,ξ /τ τ /2 2 = H s ( Ω) = B s Ψ,2 (L 2( Ω)) (equiv. norms) for all 0 s < min{3/2, s Ω }, e.g. L 2 ( Ω) = B 0 Ψ,2 (L 2( Ω))

14 typical results on (complex) interpolation typical characterization of standard embeddings B s Ψ,q 0 (L p0 ( Ω)) B α Ψ,q (L p ( Ω)) (possible only if s α) and results on their best n-term wavelet approximation (rate: γ < [s α]/2) under certain conditions on the parameters: α B s Ψ,p(L p ( Ω)) X k ϱ ( Ω) B α Ψ,τ (L τ ( Ω)), α s < α < 2s 2s s Ω s s H s B s Ψ,p (L p) α s H s B s Ψ,p (L p) B α Ψ,τ (L τ ) α B α Ψ,τ (L τ ) L 2 2 τ p p L 2 2 p τ p

15 for the double layer equation Theorem (Dahlke, W. 203) Let s (0, ), k N, and ϱ (0, min{ϱ 0, k}) for some ϱ 0 = ϱ 0 ( Ω) (, 3/2). Moreover let α and τ be given s.t. τ = α and 0 α < 2 min{ϱ, k ϱ, s} and assume that Ψ Ω has d k (int.) vanishing moments. Then for every RHS g H s ( Ω) X k ϱ ( Ω) the double layer eq. has a unique solution u B α Ψ,τ (L τ ( Ω)). Furthermore, if s [0, s) then ) σ n (u; H s ( Ω) n γ 0 γ < ] [ s min{ϱ, k ϱ, s}. s

16 Assume for simplicity that min{ϱ, k ϱ} s. Then we can take α = 2s δ and γ = [s s ] δ (for all δ > 0 small). Conclusion Best n-term approximation rate γ (benchmark for adaptive schemes) is twice as large as the rate for uniform approx.! 2s α θ α B α θ Ψ,p θ (L pθ ) B α Ψ,τ (L τ ) s H s s H s L 2 2 p θ τ p

17 We introduced new Besov-type spaces B α Ψ,q (L p( Ω)) on patchwise smooth manifolds... recovered many typical properties... derived a regularity/approximation assertion for some boundary integral equation (double layer eq.) which is of fundamental relevance in practice Work in progress: dependence on the chosen basis Ψ = (Ψ Ω, Ψ Ω ) other important (e.g. single layer) application to real-life problems (Helmholtz eq.) ( )

18 Dahlke, W.: on patchwise smooth manifolds. submitted (203). (Preprint: Bericht Mathematik Nr des Fachbereichs Mathematik und Informatik, Universität Marburg) Dahmen, Schneider: Composite wavelet bases. Math. Comp. 68, (999). Elschner: The double layer potential operator over polyhedral domains I. App. Anal. 45, 7 34 (992). Sauter, Schwab: Boundary Element Methods. Springer, Berlin, 20. Triebel: Theory of Function Spaces III. Birkhäuser, Basel, 2006.

19 Dahlke, W.: on patchwise smooth manifolds. submitted (203). (Preprint: Bericht Mathematik Nr des Fachbereichs Mathematik und Informatik, Universität Marburg) Dahmen, Schneider: Composite wavelet bases. Math. Comp. 68, (999). Elschner: The double layer potential operator over polyhedral domains I. App. Anal. 45, 7 34 (992). Sauter, Schwab: Boundary Element Methods. Springer, Berlin, 20. Triebel: Theory of Function Spaces III. Birkhäuser, Basel, Thank you!