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
Outline Singularities in BVP Construction of Graded Mesh FE-Scheme Error Estimates Numerical Results
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.
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.
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.
Outline Singularities in BVP Construction of Graded Mesh FE-Scheme Error Estimates Numerical Results
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
Example : 1D Graded Mesh 1D graded mesh 1 0.8 0.6 µ 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r Figure : 1D graded mesh with varying µ
Example : L-Shape Graded Mesh graded mesh with µ = 0.4 No grading with µ = 1.0
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
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).
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).
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
Outline Singularities in BVP Construction of Graded Mesh FE-Scheme Error Estimates Numerical Results
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
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,
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,
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
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
Outline Singularities in BVP Construction of Graded Mesh FE-Scheme Error Estimates Numerical Results
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.
Gradient of the solution. Ref: [http://www.math.uci.edu/chenlong/226/ch4afem.pdf]
Knot Vector Grading: insert knots closer to singularity. 10 1 10 2 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 10 3 10 4 10 5 10 0 10 1 10 2 10 3 10 4 DOFs Figure : L 2 -Errors of graded mesh plotted against DOFs.
10 3 10 2 10 1 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 10 0 10 1 10 2 10 0 10 1 10 2 10 3 10 4 DOFs Figure : H 1 -Errors of graded mesh plotted against DOFs.
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.