Mesh Grading towards Singular Points Seminar : Elliptic Problems on Non-smooth Domain

Transcription

1 Mesh Grading towards Singular Points Seminar : Elliptic Problems on Non-smooth Domain Stephen Edward Moore Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Linz, Austria Numerical Analysis Seminar, JKU, January 14, 2014

2 Outline Singularities in BVP Construction of Graded Mesh FE-Scheme Error Estimates Numerical Results

3 Singularities in BVP Lu ( ) u u a ij +b i +σu = f, in Ω x i x j x j ( ) u u S1 = 0, a ij N +σu S2 = 0 Cases of Singularities: Discontinuous coefficients (a ij ). Discontinuous right-hand side (f). Jump in boundary data. Domains with corner points.

4 Domains with corner points x 2 = y x2 = y Ω β Ω ω x 1 = x Ω ω β Ω x1 = x Heart L-shape where β > π. Singularity at ω.

5 Recap... u +u = f inω R 2 u n = 0on Ω := S Find u H = W 1 2 (Ω) : L Ω (u,φ) = (f,φ) Ω, φ H where u = u R +u S ur Regular part. us = 0<λ<1γψ(r,θ) is Singular part. ψ(r,θ) = ξ(r)r λ cosλθ λ = π/β and 1/2 < λ < 1. ξ(r) smooth cut-off function.

6 Outline Singularities in BVP Construction of Graded Mesh FE-Scheme Error Estimates Numerical Results

7 Construction of Graded Mesh Aim : To construct a mesh such that node distribution becomes denser towards the singular point. x 2 = y let h > 0 r i = (ih) 1/µ, i = 0,...,N ω r x 1 = x h i = r i r i 1, i = 1,...,N S = Ω Ω β with 0 < µ < 1 and N = [h 1 ]. C 1 (ih) µ 1 1 h i h C 2(ih) µ 1 1

8 Example : 1D Graded Mesh 1D graded mesh µ r Figure : 1D graded mesh with varying µ

9 Example : L-Shape Graded Mesh graded mesh with µ = 0.4 No grading with µ = 1.0

10 Example : Curved domain ϑ i = {r i 1 r r i }, i = 1,...,N Domain with polygon x2 = y D = N i=1 ϑ i S h = Ω h ex Ω h ex ω Ω Ω x1 = x S h is located exterior to Ω (S h Ω = ) nodes x S h z D : dist(x,s) δh 2 = O(h 2 ) nodes x S h z ϑ i : dist(x,s) δh 2 i β

11 S h = N i=1 lh j nodes (l h j ) ϑ i : l h j l 0h i l h j D = : l h j l 0h where l 0,δ C(h). node description l h j Triangulation (T h ) of Ω h ex : Conditions to satisfy: 1. δ,δ T h : δ δ := {,joint vertex,joint edge} δ ϑi : l 1 h i l = e l 2 h i, e δ δ D = : l1 h l = e l 2 h, e δ 2. θ δ θ 0 = const. > 0 δ T h 3. N i := {δ T h : δ ϑ i } N 0 i where l 1,l 2,N 0 C(h).

12 S = Ω S = Ω S h = Ω h ex S h = Ω h ex Ω Ω h h Ω h ex Ω h ex Figure : dist(s,s h ) = O(h 2 ) Figure : dist(s,s h ) = O(h) bad condition number in the case of natural BC (right figure).

13 Remark R h := number of nodes = O(h 2 ). number of nodes not located in D = O(h 2 ) number of nodes (M) located in D is : M C N N N i CN 0 i Ch 2 = O(h 2 ). i=1 i=1

14 Outline Singularities in BVP Construction of Graded Mesh FE-Scheme Error Estimates Numerical Results

15 FE-Scheme Error Estimates x2 = y Ω Find u W 1 2 (Ω) : L Ω (u,φ) = (f,φ) Ω φ H S h = Ω h ex Ω Ω h ex ω β x1 = x Find ṽ H h = S 1 (Ω h ex) : L Ω (ṽ,φ) = (f,φ) Ω φ H h

16 Cea s Lemma : u ṽ 1,Ω C min φ Hh u φ 1,Ω u = γψ +w H with w W2 2(Ω) W2 2 ( Ω) Interpolant : Π h : L 2 H h ũ = Π h u = γπ h ψ +Π h w = γ ψ + w H h u ṽ 1,Ω C u ũ 1,Ω h ex For δ = T h : ψ = 0,i.e.u = w u ũ 2 1, = w w 2 1, Ch2 w 2 2,

17 CASE I : [ϑ 1 ϑ 2 ] : Show u ũ 2 1, C (h 2λ/µ γ 2 +h 2 w 2 2 ) Ideas for Proof: 1. r Ch 1/µ 2. ψ 2 1, Ch2λ/µ and ψ 2 1, Ch2λ/µ. 3. w w 2 1, Ch2 w 2 2, 4. u ũ 2 1, 4 γ 2 ( ψ 2 1, + ψ 2 1, ) +2 w w 2 1,

18 CASE II : [ϑ 1 ϑ 2 ] = : [ ] N u ũ 2 1,Ω C h 2λ µ γ (1+ 2 i 2λ h µ )+h 4 2 w 2 ex 2,Ω i=1 Ideas for Proof: 1. D 2 ψ 2 Cr 2λ 4 2. w w 2 1, Ch2 w 2 2, ( ) 3. u ũ 2 1, C hi 4r2λ 4 i γ 2 +hi 2 w 2 2, 4. sum over all triangles T h

19 For µ = 1 N i 2λ 4 C <, N. i=1 u ṽ 1,Ω h ex Ch λ f 0,Ω For µ < λ, i.e. λ µ > 1 N i=1 i 2λ µ 4 Ch 2 2λ µ u ṽ 1,Ω h ex Ch f 0,Ω

20 Outline Singularities in BVP Construction of Graded Mesh FE-Scheme Error Estimates Numerical Results

21 Construction of Graded Mesh: IGA Example : L-Shape with 2 patches { u = 0 in Ω u = g D on Ω u(x,y) = (x 2 + y 2 ) 1/3 sin((2arctan(y/x)+ π)/3). u u h L 2 = O(h 4/3 ). u u h H 1 = O(h 2/3 ). Knot Vector Ξ 1,2 = {0,0,1,1}. p = 1 (bilinear FEM). Figure : L-Shape solution on multi-patch.

22 Gradient of the solution. Ref: [

23 Knot Vector Grading: insert knots closer to singularity L 2 Error Estimate for Graded Mesh versus DOFs µ = 0.2 µ = 0.4 µ = 0.6 µ = 0.8 µ = 1.0 Errors DOFs Figure : L 2 -Errors of graded mesh plotted against DOFs.

24 µ = 0.2 µ = 0.4 µ = 0.6 µ = 0.8 µ = 1.0 H 1 Error Estimate for Graded Mesh versus DOFs Errors DOFs Figure : H 1 -Errors of graded mesh plotted against DOFs.

25 Conclusion 1. For µ = 1, the convergence rate for p 1 is u uh L 2 = O(h 4/3 ). u uh H 1 = O(h 2/3 ). 2. Knot grading with 0 < µ < 0.6 u uh L 2 = O(h 2 ). u uh H 1 = O(h). 3. Remarks on the artifacts in errors: condition number of matrix deteriorates for µ 0. related solver issues. computation of the H 1 norm close to the singularity.